Florida Regional Geometry Individual Test Past Paper - January 2021 PDF
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2021
Florida Regional Competition
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This is a past paper for a geometry individual test from the Florida Regional Competition in January 2021. The paper includes 28 geometry questions testing various concepts. The questions cover topics like angles, triangles, polygons, and similar figures, likely designed for high school students.
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GEOMETRY INDIVIDUAL TEST January 2021 Florida Regional Competition “NOTA” is an abbreviation for “None of the Above” and should be chosen if choices A, B, C and D are not correct. Diagrams may not be drawn to scale. All angle...
GEOMETRY INDIVIDUAL TEST January 2021 Florida Regional Competition “NOTA” is an abbreviation for “None of the Above” and should be chosen if choices A, B, C and D are not correct. Diagrams may not be drawn to scale. All angle measures are in degrees. If the question refers to an angle of a polygon, unless otherwise stated, assume it is an interior angle. !!!" 1 1. UR bisects ÐNUT. mÐNUR = (10 + mÐNUT ). Find mÐNUR. 3 o o A. 5 B. 7 C. 8o D. 10o E. NOTA 2. DWHY is isosceles with WH @ YH. If mÐY = 40o then find mÐH. A. 40 o B. 58 o C. 70 o D. 100 o E. NOTA 3. Angle R has measure forty-two more degrees than twice the measure of its complement. Find the degree measure of the supplement of ÐR. A. 106 o B. 104 o C. 96 o D. 84 o E. NOTA 4. A triangle has angle measures in the ratio of 2:3:7. Find the measure of the largest (interior) angle of the triangle. A. 120 o B. 105 o C. 100 o D. 84 o E. NOTA 5. Q, S and T are collinear mÐRSQ = (4 x + 8 y )o , mÐR = (30 y )o and mÐRTS = (2 x - 26)o. If x = 6 y + 2 then find the value of x. R Q S T A. 18 B. 20 C. 21 D. 22 E. NOTA page 1 GEOMETRY INDIVIDUAL TEST January 2021 Florida Regional Competition 6. Rectangle RSTU has RU on the y-axis and UT on the R S x-axis as shown. RU=6 and UT=8. A smaller rectangle has two !""#sides contained on the axes also, and one vertex 6 is on RT. If the vertical side of the small rectangle has length 4, then give the perimeter of the smaller rectangle. U 8 T 56 40 A. B. 13 C. D. 10 E. NOTA 3 3 7. DRST is scalene and mÐT has the least of the three angle measures of DRST. DRST ~ DWHY. Which of the following is not necessarily true? A. DWHY is scalene. B. mÐY < mÐS. C. WH < WY + HY. D. mÐY < mÐW + mÐH. E. NOTA 8. DRST ~ DMNQ. RS = ST = 7 and RT = 4. MN = a and MQ = (22 - a). Find the length of the longest altitude of DMNQ. A. 6 5 B. 2 14 C. 14 D. 2 33 E. NOTA 9. A rigid vertical pole is in a pond as shown, with its base on the pond bottom. The pole shows 10 inches above the surface of the water. Ralph (R) stands on shore and pulls the top of the pole toward him with a fishing line, as the pole’s bottom stays in the same location. The pole remains straight. When the top of the pole is 21 inches horizontally from where it originally was, the top disappears under water. How deep is the pond in inches, where the pole’s base is? Disregard the pole’s width. R 10” lake surface lake bottom 341 A. B. 341 C. 341 D. 34.1 E. NOTA 20 10. Isosceles DRST has one angle measure that is x o and another angle measure that is (2 x + 40)o. Which could NOT be the measure of one angle of DRST ? A. 20 o B. 35 o C. 70 o D. 80 o E. NOTA page 2 GEOMETRY INDIVIDUAL TEST January 2021 Florida Regional Competition 11. A regular hexagon has one angle that measures (4n)o. Find the measure of one exterior angle of a regular polygon with n sides. A. 12o B. 35o C. 45o D. 60o E. NOTA R S 12. T U V W !""# !""# RS is parallel to VW and both are perpendicular to the line that contains points T and U as shown. mÐSRT = 30o and mÐUVW = 40o. Find mÐRTU. A. 110 o B. 120 o C. 130 o D. 140 o E. NOTA 13. DRST has RS = 6, ST = 8 and RT = 10. DPQN has PQ = 8, QN = 6 and mÐQ = 90o. Which is true? A. DRST @ DPQN B. DRST @ NQP C. DRST @ DPNQ D. DRST @ DQNP E. NOTA 14. RTVU is a rectangle. S is on side RT as shown. US ^ SV , US=5 and SV=12. Find the length of ST. R S T U V 25 25 144 A. B. C. 13 - 3 2 D. E. NOTA 13 26 13 15. A right triangle has hypotenuse of length 7 and the altitude to the hypotenuse has length 2 3. The altitude to the hypotenuse intersects the hypotenuse at point P. P divides the hypotenuse into what two lengths (which total 7)? A. 1 and 6 B. 2 and 5 C. 3 and 4 D. 2 3 and 7 - 2 3 E. NOTA page 3 GEOMETRY INDIVIDUAL TEST January 2021 Florida Regional Competition !!!" 1 16. RT bisects ÐARE. mÐART = ( mÐTRE ) + 20. Find mÐARE. 3 o o A. 60 B. 40 C. 28 o D. 24 o E. NOTA 17. In DRST , A, B and C lie on the sides as shown. AT , RC and SB are medians of the triangle which intersect at point P. If PB=1.5 and PT=2.6 and the perimeter of DASP is an integer, then which could be the perimeter of DASP ? R A B P S T C A. 5 B. 6 C. 8 D. 9 E. NOTA 18. A regular polygon has one exterior angle of measure d o and one interior angle of measure 3d o. What is the sum of the interior angles of this polygon? A. 1080 o B. 900 o C. 720 o D. 540 o E. NOTA 19. A vertical flagpole is 36 ft tall and it casts a 48 ft shadow at the same time of day that a vertical tower casts a 320 ft shadow. Shadows are on the horizontal ground and perpendicular to the structures. Find the height of the tower in feet. A. 300 B. 290 C. 280 D. 240 E. NOTA 20. In right DRST , P is on hypotenuse RT and M and N are on ST so that PM=PN=NT. If mÐR = 82o then find mÐRPM. R 82 P S M N T A. 86o B. 48o C. 24o D. 22o E. NOTA page 4 GEOMETRY INDIVIDUAL TEST January 2021 Florida Regional Competition 21. Isosceles trapezoid RSTU has smaller base with length 10 and one leg with length 5. If the perimeter of RSTU is 36 then find the length of the diagonal of RSTU. A. 4 26 B. 2 29 C. 5 5 D. 185 E. NOTA 22. PQ ! MN ! ST. PR: MP: SM = 1: 2: 2.The perimeter of DRMN is 30. Find the perimeter of DRST. R P Q M N S T A. 60 B. 55 C. 50 D. 47.5 E. NOTA 23. F G 160 160 NOT M H drawn to 150 N 150 scale! 80 160 P L 60 I S 160 95 Q K J 140 R In the diagram above, octagon FGHIJKLM has angle measures 160o ,160o ,150o , ( n + 40)o , 160o , (n + 40)o , 140o and 150o clockwise from F. Points S, L and N are collinear, and points H, P, I and Q are collinear. Pentagon NPQRS intersects the octagon at L and P (which is on side HI. The pentagon has angle measures 160o , no ,95o , 140o ,and ( x - 40)o clockwise from N. Find the value of x. A. 145 B. 105 C. 95 D. 85 E. NOTA page 5 GEOMETRY INDIVIDUAL TEST January 2021 Florida Regional Competition R 24. P 6 S 2 Q 6 T S. Q and T are collinear and P is on RT. Right triangles RST and PTQ are shown with right angles at S and P respectively. RS=QT=6 and SQ=2. Find the length of PQ. 18 16 A. 4 B. C. D. 3 E. NOTA 5 5 P 25. U R Q S T !""# !""# !""# Lines RU and ST intersect line PT at Q and T respectively. mÐPQR = (a + 40)o and o æ1 ö mÐTQU = (2a - 20)o and mÐSTQ = ç a + 70 ÷. Which is true? è2 ø !""# !""# A. RU and ST are not parallel and a cannot be determined. !""# !""# B. RU and ST are not parallel and a can be determined. !""# !""# C. RU and ST are parallel and a can be determined. !""# !""# D. RU and ST are parallel but a can have more than one value. E. NOTA 26. Right triangle RST has hypotenuse RT and RS=6. R 3 If sin ( ÐT ) = then find the length of ST. 4 6 A. 10 B. 3 5 C. 3 2 S T D. 2 7 E. NOTA page 6 GEOMETRY INDIVIDUAL TEST January 2021 Florida Regional Competition 27. O is P Q the origin O S R 3 Square PQRS has vertex P on the line with equation y = x and SR on the x-axis. R has 5 a coordinates (10, 0). The length of OP is c in simplest radical form with positive b integers a, b and c. For a and b relatively prime, give a + b + c.. A. 43 B. 38 C. 34 D. 17 E. NOTA 28. Square RSTU has diagonal length 10. What is the perimeter of RSTU? A. 40 2 B. 20 2 C. 20 D. 20 E. NOTA 29. The altitude of an equilateral triangle has length 12. Give the perimeter of the triangle. A. 52 3 B. 48 3 C. 24 3 D. 18 3 E. NOTA 30. R P Q 6 S 8 T In right triangle RST with S the right angle, RS=6, ST=8, SP bisects ÐS and SQ is the median to side RT. R, P, Q and T are collinear. Find the length of PQ. 1 24 5 A. B. 5 - 3 2 C. D. E. NOTA 5 35 7 page 7 Geometry Individual 2018 January Regional “NOTA" stands for “none of these answers." Figures are not drawn to scale. 1. If Kyle does not do his homework, then he is lazy. Kyle is lazy. Which of the following must be true? a) Kyle never does his homework b) Kyle always does his homework c) If Kyle does his homework, then he is not lazy d) If Kyle is not lazy, then he does his homework e) NOTA 2. If 𝛥ABC has m∠A=20o and m∠B=40o, then what is the degree measure of ∠C? a) 30o b) 40o c) 60o d) 120o e) NOTA 3. What is the perimeter in units of an equilateral triangle with side length 2? a) b) 3 c) 6 d) 8 e) NOTA 4. In 𝛥ABC, AC=BC=4 and AB=1. E is a point on segment. If P is the maximum value of BE+AE and Q is the minimum value of BE+AE, then what is P-Q? a) 1 b) 3 c) 5 d) 7 e) NOTA 5. Find the perimeter in units of a square inscribed in a triangle with side lengths 5, 5, and 6 if one side of the square lies on the side of the triangle with length 6. a) b) c) d) e) NOTA 6. Find the sum of the coordinates of the centroid of a triangle with vertices at (4, ), (5, ), and (6,1). a) 4 b) 5 c) 6 d) 7 e) NOTA 1 Geometry Individual 2018 January Regional 7. Refer to the diagram below. In equilateral triangle 𝛥ABC with side length 3 inches, points D and E trisect , G is on , || , and ||. Find the perimeter of quadrilateral ADGF in inches. a) 3 b) 4 c) 5 d) 6 e) NOTA 8. How many sides does a dodecagon have? a) 10 b) 12 c) 14 d) 20 e) NOTA 9. Rectangle ABCD has side lengths 8 and 15 units. A right triangle with a right angle at point B and its legs extending along and is constructed such that point D bisects its hypotenuse. Find the length, in units, of the right triangle's hypotenuse. a) 32 b) 34 c) 36 d) 38 e) NOTA 10. Consider the following diagram of unit square ABCD. Point E is on , point G is on , and AE=. Point H on is projected onto at point F, and FG=AE. H is also projected onto at point I. Find the value of IH HF. 2 Geometry Individual 2018 January Regional a) b) c) d) 2 e) NOTA 11. In 𝛥ABC, AB=1, AC=3, and BC is an integer. The angle bisector through C intersects at point E. CE can be expressed in the simplest radical form for a,b Z+. Find the value of a+b. a) 32 b) 35 c) 37 d) 40 e) NOTA 12. In 𝛥ABC, the angle bisector of ∠A intersects at a right angle. If AC=4 and m∠B=45o, then the perimeter of 𝛥ABC can be written in the simplest radical form a+b for a,b,c Z+. Find the value of a + b + c. a) 4 b) 8 c) 14 d) 16 e) NOTA 13. Find the sum of all k such that the line through points (7,k) and (2,0) is parallel to the line through points (k-2,3) and (2k,6). a) -2 b) 2 c) -8 d) 8 e) NOTA 14. Refer to the diagram below. A quadrilateral is formed by intersecting unit square ABCD with unit equilateral triangle EFG. Point B lies on the centroid of 𝛥EFG and. intersects with at point Y. Find the length of segment. 3 Geometry Individual 2018 January Regional a) b) c) d) e) NOTA 15. Which of the following statements is always true regarding parallel lines and a transversal? a) Vertical angles are supplementary. b) Consecutive interior angles are congruent. c) Alternate interior angles are congruent. d) Corresponding angles are complementary. e) NOTA 16. Right triangle 𝛥ABC has a right angle at vertex A and AC>AB. Point D is the foot of the altitude from vertex A. If CD-BD=1, then find the value of AC2-AB2 in terms of BC. a) b) c) BC d) BC e) NOTA 17. Which of the following is a polygon with 10 sides? a) dodecahedron b) decagon c) dodecagon d) decahedron e) NOTA 18. Which of the following methods cannot be directly used to prove two triangles are similar? a) SSS b) AA c) SAS d) SSA e) NOTA 19. How many diagonals are in a convex 60-gon? 4 Geometry Individual 2018 January Regional a) 172 b) 1710 c) 1770 d) 10440 e) NOTA 20. A convex n-gon is divided into only triangles by segments connecting (not necessarily all) its vertices. Express the fewest number of triangles that can be formed in terms of n. a) n-2 b) n-1 c) n d) n+1 e) NOTA 21. A triangle has side lengths 3x+2, x+1, and 2x+3 for real x. Which of the following is the domain of x? a) all reals b) reals strictly greater than -1 c) reals greater than or equal to -1 d) positive reals e) NOTA 22. In right triangle 𝛥ABC, the altitude dropped from A to hypotenuse intersects at point D. Given that BD=5 and CD=4, what is the length in units of ? a) 4.5 b) 10 c) 2 d) 20 e) NOTA 23. A 30-60-90 right triangle with hypotenuse 1 has an equilateral triangle constructed outwards from each side. Find the side length in units of the equilateral triangle formed by the centroids of the three constructed equilateral triangles. a) 1 b) c) d) e) NOTA 24. Tianbo is on a hoverpad moving straight ahead towards a skyscraper at 10 meters per second, and his eye level is 10 meters above ground. If he spots the base of a skyscraper at an angle of depression of 45o, then how long from the time the angle of depression is 45o will it take for him to reach the skyscraper? a) 1 second b) 10 seconds c) 100 seconds d) 1000 seconds e) NOTA 5 Geometry Individual 2018 January Regional 25. What is formed when three or more coplanar, nonintersecting line segments of different lengths are connected at their ends to form a closed figure? a) regular polygon b) rhombus c) scalene triangle d) irregular polygon e) NOTA 26. In trapezoid ABCD, m∠C=m∠D=60o. Point E is on segment such that m∠AEB=30o. A line through D perpendicular to meets at point G. Given that BG=1 and CD=2 , find the length of. a) 3- b) 1+ c) 3+ d) 1+2 e) NOTA 27. Find the degree measure of one interior angle of a regular convex hexagon. a) 90o b) 110o c) 120o d) 150o e) NOTA 6 Geometry Individual 2018 January Regional 28. In 𝛥ABC, AC=6 and BC=5. Let M be the midpoint of. Given that is perpendicular to the angle bisector of ∠BAC, then find the perimeter in units of 𝛥ABC. a) 13 b) 14 c) 15 d) 16 e) NOTA 29. If p q is always true and q r is always true, then which of the following statements must apply? a) p r is always true b) p r is sometimes true c) p r is never true d) r p is always true e) NOTA 30. Find the maximum height (distance between two parallel sides) in units of a rhombus with side length 1. a) b) c) 1 d) e) NOTA 7 Geometry Individual January Regional For all questions, E is NOTA, which denotes that “none of the above” is correct. All diagrams are not drawn to scale. All angles are in degrees. 1. Given a regular polygon with 20 diagonals, find the measure of one interior angle. A) 8 B) 45 C) 135 D) 162 E) NOTA 2. “If I can fly, then I am superman.” Find the converse of the contrapositive of the converse of the inverse of the contrapositive of the conditional statement above. A) “If I am not superman, then I can’t fly” B) “If I am superman, then I can fly” C) “If I can fly, then I am superman” D) “If I can’t fly, then I am not superman” E) NOTA 3. In right triangle LIN , find the length of IK. A) 15 B) 5 C) 3 3 D) 34 E) NOTA 4. The supplement of an angle is thirty less than three times the angle. What is the measure (in degrees) of half the angle? 105 75 A) B) C) 52.5 D) 105 E) NOTA 4 2 5. A right triangle has one leg with a length of 6 feet, and the hypotenuse has a length of 10 feet. What is the length of the other leg, in inches? A) 8 B) 2 34 C) 72 D) 96 E) NOTA Page 1 of 6 Geometry Individual January Regional 6. Find the acute angle between the minute and hour hand at 5:19 PM. Pick the closest answer in degrees. A) 30 B) 36 C) 45.5 D) 314.5 E) NOTA 7. Given that lines v and w are parallel, lines m and n are parallel, and lines a and b are parallel, what’s the measure of x? A) 42 B) 58 C) 68 D)112 E) NOTA 8. In the concave equilateral decagon shown, find the measure of angle B. ! A) 18 B B) 36 C) 72 L D) 108 G E) NOTA N I 9. If lines a and b are parallel, find the measure of x. A) 26 B) 60 C) 110 D) 136 E) NOTA Page 2 of 6 Geometry Individual January Regional 10. In triangle KIN , Find the length of IN. 25 A) 13 B) 3 89 C) 13 114 D) 13 E) NOTA 11. Two sides of an isosceles triangle have lengths of 7 and 24. Which of the following could be the length of the third side? A) 7 B) 24 C) 25 D) 527 E) NOTA 12. Two lampposts are both at a 90-degree angle to the horizontal ground. One lamppost is 40 feet tall and the other one is 60 feet tall. Two wires go from the top of each lamppost to the bottom of the other lamppost. At which height do the wires intersect? The thickness of the wires is negligible. A) 20 B) 24 C) 30 D) 48 E) NOTA 13. Find AE when AB=6 and DE=3. Angle B and Angle ADC are right angles. A) 3 2 B) 3 5 C) 57 D) 3 7 E) NOTA 14. Triangle HEN with ST that is parallel to EN. Find the length of HS. 28 20 A) B) 3 C) 5 7 20 D) E) NOTA 3 Page 3 of 6 Geometry Individual January Regional 15. Find A+B, where A is the geometric mean of 20 and 30, and B is the arithmetic mean of 20 and 30. A) 600 B) 10 6 C) 25 + 10 6 D) 50 E) NOTA 16. Find BD when AB=7 and AC=12. 95 49 7 95 A) B) C) 5 D) E) NOTA 12 12 12 17. Timmy and Jimmy both go to the store to buy different items, and they start at the same point at the entrance. Timmy runs 45 meters north into the store for a hat. Then Timmy goes 15 meters west to reach the food court. Jimmy tries to catch up and goes 10 meters east, 20 meters north, and 5 meters west. In meters, how far does Jimmy have to walk to reach Timmy? (Timmy and Jimmy can walk through walls) A) 25 B) 5 29 C) 30 D) 5 41 E) NOTA 18. In right triangle ABC, point D is on AC. BD has a length of 20, AB has a length of 16, and BC has a length of 34. Find CD. A) 12 B) 16 C) 18 D) 30 E) NOTA Page 4 of 6 Geometry Individual January Regional 19. In the figure below, ME has points S, Q, U, A, and R on it. The numbers on each square represent the area of that square. If ME = 65, find MS. A) 9 B) 10 C) 11 D) 12 E) NOTA 20. An equilateral triangle has an altitude of 8 inches. What is the length of a side? A) 8 3 B) 4 2 C) 8 D) 16 3 E) NOTA 3 3 21. In right triangle ABC, B is a right angle, points E and D are on AC. In this triangle, BE is an altitude to AC , and BD is a median to the hypotenuse AC. Find the length of ED. 7 A) D) 4 3 5 B) 2 E) NOTA 18 C) 5 22. The property that shows IG ≅ IG is called the _______________. A) Reflexive property B) Transitive property C) Symmetric property D) Congruence property E) NOTA 23. Who was the famous Greek mathematician who discovered that the ratio of the volumes (When the sphere is inscribed inside the cylinder) of a sphere to the cylinder is 2:3? A) Archimedes B) Euclid C) Eratosthenes D) Euler E) NOTA Page 5 of 6 Geometry Individual January Regional 24. In right triangle IKN, find sin(θ). 3 4 A) 3 B) C) 5 4 5 4 D) E) NOTA 3 25. Josh is 6 feet tall and is standing upright and looking straight ahead while standing on top of and at the edge of a 39-foot tall building. He looks down at a rock on the ground that is 45 3 feet away from the base of the building. Approximately how many degrees did Josh have to look down to see the rock (angle of depression)? The width of Josh and the rock is negligible. A) 30 B) 45 C) 60 D) 90 E) NOTA 26. Find the sum of all the possible integral values of the third side of a triangle when two of the sides have lengths of 6 and 9. A) 90 B) 99 C) 105 D) 117 E) NOTA 27. A triangle has points (0,0), (11, 31 ), and (18, 2 31 ). Find the perimeter of the triangle. A) 4 5 B) 8 7 C) 12 7 D) 16 7 E) NOTA 28. If two vertical angles are also complementary to each other, then what is the supplement of one of the angles (in degrees)? A) 45 B) 90 C) 135 D) 180 E) NOTA 29. Which of the following are also all parallelograms? A) Square, Trapezoid, Kite B) Kite, Rectangle, Trapezoid C) Trapezoid, Square, Rhombus D) Rhombus, Square, Rectangle E) NOTA 30. A triangle with sides of lengths 7, 10, and 12 is which of the following? A) Acute B) Obtuse C) Right D) Isosceles E) NOTA Page 6 of 6 January Regional Geometry Individual Test For all problems, NOTA stands for “None of the Above”. 1. If two angles of a triangle measure 40◦ and 80◦ , what is the measure of the other angle of the triangle? (A) 40◦ (B) 60◦ (C) 80◦ (D) Cannot be determined (E) NOTA 2. Lines `1 and `2 intersect. If the acute angle between the two lines measures 70◦ , what is the measure of the obtuse angle between the two lines? `1 70◦ `2 (A) 110◦ (B) 140◦ (C) 160◦ (D) 290◦ (E) NOTA 3. Lines `1 and `2 are coplanar and distinct, and both are perpendicular to line `3. Which of the following must be true? (A) `1 || `2 (B) `1 ⊥ `2 (C) `1 and `2 intersect, but are not perpendicular (D) `1 and `2 are skew (E) NOTA 4. Distinct points A, B, and C lie on a line in that order, so that AB : BC = 3 : 5. What is AB : AC? (A) 3 : 8 (B) 2 : 5 (C) 1 : 2 (D) 5 : 8 (E) NOTA 5. A store has an advertisement in its window stating “If it’s hot, it’s here!”. Assuming that this statement is true (and ignoring physical impossibilities), what can we correctly conclude? (A) All items in the store are hot. (B) Any item not in the store is not hot. (C) All items are hot. (D) No item is hot. (E) NOTA 6. Which of the following is NOT a method used to prove two triangles are congruent? (A) AAS (B) ASA (C) SAS (D) SSS (E) NOTA 1 January Regional Geometry Individual Test 7. Two angles of a triangle measure 35◦ and 110◦. The triangle can be described as (A) isosceles, acute (B) isosceles, obtuse (C) scalene, acute (D) scalene, obtuse (E) NOTA 8. In triangle ABC, m∠A = 120◦. Suppose ∠B measures x degrees. Which of the following describes all possible values of x? (A) 0 < x < 60 (B) 0 ≤ x < 60 (C) 0 < x ≤ 60 (D) 0 ≤ x ≤ 60 (E) NOTA 9. Distinct points A, B, and C lie on a line in that order. Point D is not on the line. Which of the following conditions will NOT tell you that m∠DBA = 90◦ ? (A) ∠DBA = ∠DBC (B) 4DAB ∼ = 4DCB (C) DA = DC and BA = BC (D) DA = DC and ∠DAB = ∠DCB (E) NOTA 10. In right triangle ABC, m∠B = 90◦ , AB = 3 · 2009, and BC = 4 · 2009. What is AC? √ (A) 5 2009 (B) 5 · 2009 (C) 25 · 2009 (D) 5 · 20092 (E) NOTA 11. A square has diagonal length d. What is the area of the square, in terms of d? √ d2 d2 d2 2 (A) 4 (B) 2 (C) 2 (D) 2d2 (E) NOTA 12. The side lengths of a triangle are all even positive integers. The largest side has length 50. What is the smallest possible perimeter of the triangle? (A) 54 (B) 100 (C) 101 (D) 102 (E) NOTA 13. If the side length of an equilateral triangle is 6, what is the length of one of its altitudes? √ √ √ (A) 3 (B) 3 2 (C) 3 3 (D) 6 3 (E) NOTA 14. Let ABCD be a convex quadrilateral. Which of the following conditions will NOT tell you that ABCD is a parallelogram? (A) AB || CD and AD || BC (B) AB || CD and AB = CD (C) AB || CD and AD = BC (D) AB = CD and AD = BC (E) NOTA 15. In convex quadrilateral ABCD, AC bisects ∠BAD, and m∠ABC = m∠ACD. If AB = 4 and AC = 6, what is AD? √ (A) 4 (B) 2 6 (C) 6 (D) 9 (E) NOTA 2 January Regional Geometry Individual Test 16. Suppose we have a convex 20-gon. One of its diagonals is drawn, dividing it into two convex polygons, one of which has m sides, the other of which has n sides. What is m + n? (A) 20 (B) 21 (C) 22 (D) Cannot be determined (E) NOTA 17. In trapezoid ABCD, AB || CD, and AC and BD intersect at E. m∠AEB = 110◦ , m∠BAC = 30◦ , and m∠DBC = 60◦. What is the measure of ∠BCD? (A) 70◦ (B) 80◦ (C) 90◦ (D) 100◦ (E) NOTA 18. Turtle has a rectangular piece of paper with a clue on it. Before she eats it, she cuts off one of the corners as shown. What is the sum of the measures of the two marked angles in the second picture? (A) 180◦ (B) 210◦ (C) 240◦ (D) 270◦ (E) NOTA 19. Two squares are in a plane. The sides of one square intersect the sides of the other square at a total of n points, where n is a finite number. What is the largest possible value of n? (A) 1 (B) 2 (C) 4 (D) 8 (E) NOTA 20. ABCD is a convex quadrilateral with AD = BC. P is a point inside ABCD such that AP = BP = CP = DP. If m∠AP B = 20◦ and m∠CP D = 80◦ , what is m∠BAD? (A) 95◦ (B) 100◦ (C) 105◦ (D) 130◦ (E) NOTA 21. How many lines of symmetry does a regular hexagon have? (A) 3 (B) 6 (C) 12 (D) infinitely many (E) NOTA 22. Let ABC be a triangle with area 12. Point D is inside triangle ABC. What is the sum of the areas of concave quadrilaterals ABCD, BCAD, and CABD? (A) 12 (B) 16 (C) 24 (D) 36 (E) NOTA 23. For how many positive integer values of n, where n ≥ 3, does an exterior angle of a regular n-gon measure more than 10◦ ? (A) 15 (B) 17 (C) 33 (D) 35 (E) NOTA 3 January Regional Geometry Individual Test Questions 24-26 deal with the following theorem. Let ABC be a triangle, and let X, Y , and Z be points on sides BC, CA, and AB, respectively. A Y Z B X C Ceva’s Theorem states that if AX, BY , and CZ intersect at one point, then AZ BX CY · · = 1. ZB XC Y A The converse of this statement is also true, and is often used to prove three lines inter- sect at one point. Three lines that intersect at one point are said to be concurrent. 24. In triangle ABC, points X, Y , and Z are on sides BC, CA, and AB, respectively, such that lines AX, BY , and CZ are concurrent. AZ = 1, BX = 3, XC = 2, CY = 4, and Y A = 3. What is ZB? (A) 1/2 (B) 8/9 (C) 9/8 (D) 2 (E) NOTA 25. In triangle ABC, points X, Y , and Z are on sides BC, CA, and AB, respectively, AZ BX CY such that AZ = ZB, BX = XC, and CY = Y A. Then ZB · XC · Y A = 1 · 1 · 1 = 1, so the converse of Ceva’s theorem says that lines AX, BY , and CZ are concurrent. This proves that which of the following intersect at one point? (A) medians (B) angle bisectors (C) perpendicular bisectors (D) altitudes (E) NOTA 26. In triangle ABC, points R and S are on sides AB and AC such that 4ARS ∼ 4ABC. Point P is on side BC such that lines AP , BS, and CR are concurrent. If AR = 3, RB = 2, and RS = 6, what is BP ? (A) 2 (B) 3 (C) 4 (D) 5 (E) NOTA 4 January Regional Geometry Individual Test 27. In right triangle ABC with hypotenuse AC, AB = 7 and AC = 10. Which of the following angles is the smallest? (A) ∠A (B) ∠B (C) ∠C (D) an angle measuring 100◦ (E) NOTA 28. In the following diagram, ABCD is a square with area 16. What is the area of the shaded region? A B ◦ 30 30◦ 30◦ 30◦ D C √ √ (A) 7 − 4 3 (B) 16 − 8 3 (C) 3 (D) 16/5 (E) NOTA 29. Let m and n be integers greater than or equal to 3, such that m > n. The interior angle measure of a regular m-gon is k/2 times the interior angle measure of a regular n-gon, where k is some positive integer. How many possible ordered pairs (m, n) are there? (A) 4 (B) 5 (C) 6 (D) 8 (E) NOTA 30. In convex hexagon ABCDEF , m∠ABC =√m∠CDE = m∠EF A = 100◦ , AB = BC = 6, CD = DE = 3, and EF = F A = 3 3. What is the measure of ∠F AB? (A) 110◦ (B) 130◦ (C) 140◦ (D) 170◦ (E) NOTA 5 January Regional Geometry Individual Test The abbreviation NOTA denotes “none of these answers”. All numbers are assumed real. Diagrams may not necessarily be drawn to scale. Good luck! 1. Given a regular polygon with 23 sides, how many diagonals does the polygon have? A. 230 B. 253 C. 276 D. 299 E. NOTA 2. What is the measure of an angle in degrees such that 3 times its compliment is equal to 3/5ths of its supplement minus ½ of itself? A. 1620/19 B. 1296/17 C. 900/11 D. 600/17 E. NOTA 3. Given the 3-4-5 triangle Δ ABC to the right, find the length of n, B the altitude drawn from point D, the intersection of the medians drawn from vertices A and B, down to AC. A. 1 5 B. √3/2 3 D C. √13/3 n D. 2 E. NOTA A 4 C 4. Which of the following will always be in the interior of the triangle? I) Incenter II) Circumcenter III) Orthocenter IV) Centroid A. IV only B. I & IV C. II & III D. I, III & IV E. NOTA 5. George, who lives in a gridded world, is hungry and wants to go get cupcakes and frozen yogurt. If he is currently at the point (0, 5) and the cupcake store is at the point (12, 7) and there is a frozen yogurt store at every location along the line y=0, what is the shortest distance he can travel to the cupcake store if he gets frozen yogurt along the way? A. 10 B. 2√37 C. 12√2 D. 13 E. NOTA 6. Given the parallelogram to the right, what is the measure of the angle labeled 2y (all measures are presented in degrees)? 4x+2y 4x A. 45 B. 55 C. 57.5 D. 60 2y z E. NOTA 7. The shortest leg of a 30-60 right triangle measures 2 ft. What is the length of the hypotenuse in inches? A. 2√3 B. 4 C. 24√3 D. 48 E. NOTA 8. Given the statement, “if your name is Steve, you like drawing giraffes with obnoxiously long necks,” what is the converse of the contrapositive of the inverse of the converse of the contrapositive of the statement? A. If your name is Steve, you like drawing giraffes with obnoxiously long necks. B. If your name is not Steve, you do not like drawing giraffes with obnoxiously long necks. C. If you like drawing giraffes with obnoxiously long necks, your name is Steve. D. If you do not like drawing giraffes with obnoxiously long necks, your name is not Steve. E. NOTA 9. What is the sum of all the degrees of the interior angles in a 23-sided convex polygon? A. 3780 B. 3960 C. 4140 D. 4500 E. NOTA January Regional Geometry Individual Test 10. Find the centroid of a triangle with vertices on the coordinates, (21, -9), (-7, 3) and (14, -6). A. (9, -4) B. (28/3, 0) C. (28/3, -4) D. (9, 4) E. NOTA 11. How many of the following are true concerning rhombuses? I) Diagonals perpendicularly intersect each other II) Diagonals bisect two angles III) Opposite angles sum to 180 degrees IV) Opposite angles are congruent V) Diagonals bisect each other VI) Opposite sides are congruent A. 3 B. 4 C. 5 D. 6 E. NOTA 12. Which of the following is not illustrated by the edges and vertices of the rectangular pyramid shown on the right? A. 3 collinear points B. 3 non-coplanar points C. A pair of parallel line segments D. A pair of skew line segments E. NOTA 13. Given that AB || CD; BC bisects ACD; the lines AC, CD and BC intersect at point C; the angle marked as 40°; w what is the value, in degrees, of w? B A. 70 A B. 75 C. 100 40° D. 110 C D E. NOTA 14. What is the name of the lines that meet at a point equidistant from all three sides of a triangle? A. Angle bisector B. Perpendicular bisector C. Altitude D. Median E. NOTA 15. Which of the following does not represent the side lengths of a possible right triangle? A. 7, 24, 25 B. 5, 12, 13 C. 20, 21, 29 D. 11, 60, 61 E. NOTA 16. Within a regular hexagon, a square is inscribed with side lengths 6. What is the length of one side of the hexagon? A. 3 + √3 B. 6 - √3 C. 2 + 3√3 D. 4 - √3 E. NOTA 17. Which of the following is not a valid theorem for proving similar triangles? A. AA B. SSA C. SAS D. SSS E. NOTA 23 18. A midsegment is drawn on a trapezoid with parallel bases with lengths 10 and 23. What is the length of the z, the center-most segment among the trisected pieces of the midsegment? z A. 6.5 B. 8 C. 11 D. 13 E. NOTA 10 January Regional Geometry Individual Test 19. In a 72-sided regular convex polygon, what is the measure (in degrees) of one of the angles? A. 160 B. 167.5 C. 170 D. 175 E. NOTA 20. Given the triangle as diagramed to the right with side lengths and angle measures as indicated, what is the length of the side marked y? A. 23/2 + 23√3/2 75° B. 46 23 C. 23√2 + 23√3/2 D. 23√3 – 23/2 E. NOTA 60° y 21. What is the cosine of an angle in an equilateral triangle? A. ½ B. √2/2 C. √3/2 D. 1 E. NOTA 22. i, j and k represent the side lengths of a right triangle with side length i as the hypotenuse, while h represents altitude drawn to the hypotenuse. What is i written in terms of j, k and h? A. (j + k)/h B. jk/h C. √(j2 + k2)/h D. 2(j + k)/h E. NOTA 23. Given that ΔCAT and ΔDOG are similar triangles and line segments CA, AT, DG and DO have respective side lengths 3, 7, 4 and 5, what is semi-perimeter of ΔDOG? A. 31/5 B. 31/3 C. 62/5 D. 62/3 E. NOTA 24. A square is inscribed inside an equilateral triangle with side lengths 2. What is the length of one side of the square? A. 3√3 – 3 B. 2√3 – 4 C. 4√3 – 6 D. 6√3 – 9 E. NOTA 25. In ΔCOW, sides CO and CW have lengths of 5 and 4 respectively. Angle OCW is bisected by line segment CD with point D lying on the line segment OW, which has length 18. What is length of segment DO? A. 7 B. 8 C. 9 D. 10 E. NOTA 26. In a right triangle, two of the sides have measures 3 and 3√2, what is the product of all possible values for the unknown side length. A. 9√2 B. 9√3 C. 27 D. 12√2 E. NOTA 27. What is the distance between the coordinates (3, 23) and (-17, 44). A. 23 B. 25 C. √823 D. 29 E. NOTA 28. What is the maximum number of sections that can be formed when a circle is divided by 23 chords? A. 230 B. 242 C. 253 D. 277 E. NOTA 29. Given an isosceles triangle Δ RAM with vertex R and a square WORM; line segments RA and AM which have lengths 3 and 4 respectively, what is the perimeter of the pentagon WORAM? A. 16 B. 17 C. 19 D. 23 E. NOTA A 30. In the diagram to the right, G and O are right angles. If lengths TO, RT and GT are 3, 5 and 23 respectively, what is the length of side GA? G A. 69/4 B. 92/3 C. 115/3 T D. 115/4 E. NOTA O R January Regional Geometry Individual For all questions, NOTA is defined as “None of the above answers is correct”. 1) What is 2 times the supplement of the complement of (2x – 5) degrees? All answers are in degrees. A. 170 + 4x B. 95 – 4x C. 4x + 90 D. 90 – 4x E. NOTA 2) A right triangle has legs and. Given this information, find the length of the hypotenuse of the right triangle. A. B. C. D. E. NOTA 3) In a 45-45-90 right triangle, the hypotenuse has length 64. Find the length of the longest leg of this triangle. A. 32 B. C. D. Not enough information E. NOTA 4) Given that, in the diagram to the right, parallel lines a and b are intersected by line c, and that the sum of angle 1 and angle 2 is 150 degrees, find the measure of angle 3 in degrees. A. 150 B. 105 C. 75 D. 30 E. NOTA 5) Given that the sum of the sides of a triangle is 24, and that the triangle has sides lengths of 9, 2x + 2, and x + 4. Find the value of x. A. 3 B. 6 C. 2.5 D. 10/3 E. NOTA 6) Find the minimum distance between the point (5, 4) and the line 5x – 2y = 3. A. B. C. D. E. NOTA 7) Find the measure of the largest angle of the octagon with angles measuring: 3x, 2x + 5, 15, 25 – 3x, 13, 7x – 65, and 6x – 23. A. 421 B. 453 C. 444 D. 518 E. NOTA January Regional Geometry Individual 8) Standing on top of a light-house, Gulete looks at a sail-boat 25 meters away from the light-house. The angle formed between Gulete’s line of vision and the light-house is 60 degrees. Find the height of the light- house, in meters, if Gulete is 3 meter tall. A. B. C. D. 25 E. NOTA 9) Find the maximum number of regions formed by the intersection of 11 lines. A. 34 B. 64 C. 67 D. 134 E. NOTA 10) Given the randomly placed point N in rectangle ABCD along with segments w, x, y, and z, all of which are drawn from the vertices of the rectangle to point N. Find the length of segment y if it is known that x = 4, w = 5, and z = 7. A. B. C. D. E. NOTA 11) Given angle bisector AD, and that side AB = 8, side AC = 11, and side BC = 16, find the length of segment BD. (refer to the diagram at the right) A. 108/19 B. 64/19 C. 168/19 D. 118/19 E. NOTA 12) Find the greatest value of x given that the distance between (x, 5) and (8, 13) is 10. A. 7 B. 2 C. 14 D. 4 E. NOTA 13) Find the sum of the interior angles of a polygon with 23 sides. A. 3780 B. 4140 C. 3600 D. 2360 E. NOTA January Regional Geometry Individual 14) Which of the following is not a property of parallelograms? A. opposite angles are congruent B. diagonals bisect each other C. adjacent angles are supplementary D. opposite sides are congruent E. NOTA 15) In triangle ABC, an altitude from point B to point D is drawn to hypotenuse AC. Given that this triangle has sides AB = 63, and BC = 84. Find the length of altitude BD drawn to the hypotenuse of this right triangle. A. 352/6 B. 252/5 C. 152/6 D. 302/6 E. NOTA 16) Find the geometric mean of 20 and 30. A. B. C. D. E. NOTA 17) 25 people meet twice a week. At the end of each meeting, only 2 people shake hands. Given that 2 people cannot shake hands with each other more than once, how many weeks will it take for all 25 people to shake hands? A. 300 B. 150 C. 600 D. 75 E. NOTA 18) Find the number of diagonals in a 25-gon. A. 275 B. 300 C. 600 D. 550 E. NOTA 19) Which of the following do not necessarily determine a plane? A. 3 non-collinear points B. A line and a point not on the line C. 2 non-parallel intersecting lines D. 2 skew lines E. NOTA 20) Two vertical angles have measures and. Find the value of x. A. -1 B. 8 C. 16 D. -4 E. NOTA 21) A triangle has side lengths 83, S, and 25. Find the greatest possible integral value for S. A. 105 B. Infinity C. 104 D. 109 E. NOTA 22) Find the length of the shortest side of a 30-60-90 triangle with hypotenuse of length 22. A. B. C. D. E. NOTA 23) Find the length of the median to the hypotenuse of a right triangle with sides of length 7, 24, and 25. A. 12 B. 3.5 C.14 D. 12.5 E. NOTA January Regional Geometry Individual 24) What is ? A. B. C. D. E. NOTA 25) From a point in the interior of an equilateral triangle, altitudes to the 3 sides are drawn. These altitudes have lengths 2, 6, and 4. Find the side length of this triangle. (Refer to the diagram at the right) A. B. C. D. E. NOTA 26) What is the sum of the squares of the medians of a triangle with side lengths 6, 8, and 10? A. 150 B. 100 C. 200 D. 75 E. NOTA 27) Which of the following do not prove lines parallel when they are intersected by a transversal? A. Alternate interior angles are congruent B. Alternate exterior angles are congruent C. consecutive angles are supplementary D. adjacent angles are supplementary E. NOTA 28) Given the diagram to the right, with segment ED parallel to BC, find the length of segment AB. A. 24/5 B. 5/4 C. 4/5 D. 5/24 E. NOTA 29) Which of the following do not prove triangles similar? A. AA B. SSS C. SSA D. SAS E. NOTA 30) Given the following trapezoid with angle B equal to 40 degrees. What is the measure of angle A in degrees? A. B. 160 C. D. 130 E. NOTA GEOMETRY INDIVIDUAL TEST January Regional Competition The abbreviation “NOTA” means “None of the Above” and should be chosen if 12 M choices A, B, C and D are not correct. ? 15 Diagrams may not be drawn to scale. Angle measures are in degrees. An exterior angle of a polygon forms a linear 4. A ladder is leaning against a vertical pair with its corresponding interior angle. wall. Mortimer (M) is on the ladder so that he is 15 feet above ground 1. SU bisects RST. If mRSU = (6 x + 8)o and 12 feet below (vertically) the top of the ladder. If the ladder is 36 feet and mRST = (10 x + 30)o then find the long, then how many feet is Mortimer, value of x. horizontally, from the wall? A. 6 B. 7 A. 4 7 B. 9 7 C. 8 D. 11 E. NOTA C. 5 15 D. 7 14 E. NOTA G 2. ladder F 9 H J K 5. A vertical wall has a ladder leaning against it, so that the foot of the ladder is b feet from the base of the wall. The top of the ladder is 9 feet above the L M N base of the wall. The ladder is (b + 1) feet P Q long. The length of the ladder is how many feet? Parallel lines HK and LN intersect A. 30 B. 32 transversals PG and FQ at points J, C. 40 D. 41 M and N as shown above. E. NOTA mFJH = ( x + 20)o , mGJK = (4 x + 2)o and mLMP = (3x + 14)o. What is the 3. A hexagon has interior angle measures measure of MJN ? 110o , (2a − 10)o , (3a − 30)o , (2a − 10)o , (3a − 30)o , and 90o. Which is NOT A. 32o B. 50o a possible measure of an exterior angle C. 84o D. 98o of the hexagon? E. NOTA A. 30o B. 50o C. 70o D. 90o E. NOTA GEOMETRY INDIVIDUAL TEST January Regional Competition 6. RST has a right angle at R. RS=10 9. A triangle has sides of lengths 3 cm, and RT=24. 4 cm and 4.5 cm. Which statement S N is on ST N must be true? so that 10 A. The triangle is obtuse. RN ⊥ ST. B. The angles are in the ratio of R 24 T 3: 4: 4.5. Find the distance C. The smallest angle of the triangle between N and T. is 45 o. 676 D. There is no such possible triangle. A. 12 3 B. E. NOTA 15 288 120 T C. D. 13 13 E. NOTA A M X S 120o N 7. Isosceles triangle RST has base RS. o P 110 If mR = (2 x + 10)o and mT = (6 x)o Q then what is the measure of one base 40o angle? R Y o 170 A. 42o B. 14 10. Points S, T, A, R and Y form the “outer 510 o points” of a non-convex decagon (star) C. D. 38o as shown above. Pentagon PXMNQ 7 has its vertices on the vertices of the E. NOTA star. mPXM = 120o , mY = 40o and mPQN = 110o. Find mS. A. 10 o B. 40 o 8. Rhombus RSTU is not a square. It has the same height as a square C. 45 o D. 50 o PQMN. Which statement below must E. NOTA be true? A. RS PQ 11. A square has perimeter 100 and B. RSTU has an angle that is 30o an equilateral triangle has the C. RT could be equal to PM. same length sides as the square. D. RSTU could have the same What is the ratio of the height of the perimeter as PQMN. triangle to the diagonal of the square? E. NOTA A. 3 : 2 2 B. 3 : 2 C. 2 3 : 2 D. 5 : 2 3 E. NOTA GEOMETRY INDIVIDUAL TEST January Regional Competition 12. An isosceles triangle has two angles R with measures (40 + 3 x)o degrees and P o (4 x). If x is an integer, which can be the value of x ? A. 40 B. 30 C. 12 D. 10 S T E. NOTA 16. RST has a right angle at S. P is on RT so that RT ⊥ SP. SR = 3 5. RP = ( x − 2) and PT = (2 x + 2). Find 13. A pentagon has two angles of equal measure, and the other angles the length of SP. have measures 140o , 100o and 90o. A. 3 B. 4 C. 5 D. 6 What is the measure of the angle E. NOTA which is the second largest interior angle? A. 100 o B. 105 o R C. 110 o D. 112.5 o E. NOTA P Q 14. In ABC the complement of A is forty M N less degrees than the measure of half of the supplement of A. The S 12 T supplement of B is five more degrees than twice the complement of B. In 17. To RST , lines PQ and MN are ABC , what is the measure of C ? drawn parallel to ST with points o o A. 100 B. 95 P and M on RS , and points Q and C. 90 o D. 80 o N are on RT. ST=12 and RS=18. E. NOTA If PR: PM: MS = 2: 3: 4, then find the ratio of PQ to MN. A. 2: 7 B. 5: 9 C. 2: 3 D. 2: 5 15. An equilateral triangle with height 12 E. NOTA has what perimeter? A. 12 3 B. 24 C. 24 3 D. 36 18. In RST , RS=12, RT=10 and ST is E. NOTA a positive integer. If RST is an obtuse triangle, how many different possible lengths exist for ST ? A. 19 B. 17 C. 11 D. 10 E. NOTA GEOMETRY INDIVIDUAL TEST January Regional Competition 19. A regular polygon has one exterior 23. Four squares are drawn, with the side angle that has measure of n degrees. length of the largest (RSTU) 32 cm. The polygon has (2n + 16) sides. What is The next square has vertices on the the measure of one interior angle of the midpoints of RSTU. Each subsequent polygon? square is drawn from the midpoints of the sides of the previous square. Find A. 120 o B. 135 o the side length of the smallest square. C. 144 o D. 170 o E. NOTA R R S U S 20. X T U T A. 8 2 cm B. 8 cm C. 4 2 cm D. 4 cm In rhombus RSTU, mSRT = 50o. RT E. NOTA and US intersect at X. Find mUXT. A. 70 o B. 80 o C. 90 o D. 100 o 24. One interior angle of a regular E. NOTA polygon has degree measure that is 40 degrees more than six times the degree measure of one exterior angle. How many sides does the polygon 21. A convex polygon has 104 diagonals. have? How many sides does it have? A. 24 B. 20 A. 16 B. 14 C. 12 D. 10 C. 18 D. 15 E. NOTA E. NOTA 22. In parallelogram RSTU, X is the intersection of the diagonals. UX = (4a + 12) , XS = (9b − 10) and RX = (2a + 30). Find the value of (9b − 4a). A. 50 B. 42 C. 22 D. 12 E. NOTA GEOMETRY INDIVIDUAL TEST January Regional Competition 25. 27. RST ~ MNP. RS = 10 , MN = 12 , Squares S and T share a vertex. mT = 20o. Find mP. o 50 o A. 24 B. C. 22o X 3 S D. cannot be determined E. NOTA S 1 1 1 1 R T 28. U 1 In the diagram above, the square T labeled X shares a vertex of square S 1 and has one side on squares T and R as 3 shown; one of its vertices is on the line 1 V which contains a side of both R and T. 1 Squares S and T have one vertical side 1 collinear, as shown. The side of square W R is 4 cm and the side S is 4 cm and the side of square T is 7 cm. Find the side The diagram above has all drawn length of the square labeled X, in cm. segments vertical or horizontal. The A. 15 B. 13.75 smallest square has side length 1 and the C. 12.5 D. 11.25 largest square has side length 4. A bug E. NOTA travels from vertex S to T to U to V to W. All of these points are vertices of at least one of the shown squares. The total distance traveled by the bug was 26. P Q a + b , for a and b positive integers. Find a + b. A. 80 B. 72 R S C. 60 D. 58 E. NOTA The regular hexagon and regular 29. RST ~ MNP. RS = 10 , MN = 12 , pentagon above share side QR. RS is ST = 17 , and RT = 26. Find the a side of the hexagon. Give the perimeter of MNP. measure of PQS. A. 63.6 B. 44.16 A. 168 o B. 150 o C. 42 D. RST is not possible E. NOTA C. 138 o D. 142 o E. NOTA 30. RST ~ MNP. RS = 10 , MN = 12 , and the area of RST is 1000. Find the area of MNP. A. 1440 B. 7200 C. 72000 D. RST is not possible E. NOTA Geometry Individual Test_ January Regional For all questions, E. NOTA means “none of the 5. The measure of a supplement of an angle is four above answers is correct”. Diagrams are NOT times the measure of the complement of the angle. drawn to scale. Find the measure of the complement. A. 30 B. 40 C. 60 1. In the diagram , given D. 80 E. NOTA m∠1 = 57°, m∠2 = (3x + 2)°, m∠3 = (4 x − 5)°. Find the degree measure of the smallest angle. 6. In the diagram, a || b, c || d , m∠1 = 3x + 20, m∠2 = 2 y − 25, m∠3 = x + 50, m∠4 = 3x + y. Find the value of x + y. A. 18° B. 56° C. 57° D. 58° E. NOTA 2. In the diagram, given m∠1 = 63° , m∠2 = (2 x + 3)°, m∠3 = 3(2 x + 3)°. A. 15 B. 35 C. 70 Find the degree measure of ∠ 2. D. 85 E. NOTA 7. Which of the following statements is FALSE? A. Two planes perpendicular to the same line at different points of the line are parallel. B. If two lines are parallel, a plane containing one and only one of them is parallel to the other line. A. 14.25° B. 18.75° C. 31.5° C. Two lines perpendicular to the same line are D. 40.5° E. NOTA parallel to each other. D. If two parallel planes are cut by a third plane, 3. Which statement is true for ALL parallelograms? then the lines of intersection are parallel. A. The diagonals are perpendicular to each other. E. NOTA B. The diagonals are congruent. C. The diagonals bisect opposite angles. 8. In the figure, AB = BC = CD = ED = EF = 1. D. The diagonals bisect each other. Find the length of AF. E. NOTA A. 2 B. 3 C. 6 D. 5 4. Quadrilateral ABCD is a parallelogram. E. NOTA Given: AB = x + 5, BC = 2 x, CD = y − 5, AD = y. Determine the length of AB. A. 0 B. 5 C. 10 D. 15 E. NOTA Geometry Individual Test_ January Regional 9. In ΔABC , E is a point on AC , D is a point on 15. Which is the longest segment in the diagram AB such that ED || CB. given AB = 2 x, EC = 5, AD = 6, BD = x + 7, and ED = 7. m∠D = 90°, m∠CAD = 60°, m∠CAB = 34°, m∠ACB = 86°. Find the perimeter of ΔABC. 1 1 5 A. 14 B. 23 C. 54 2 3 6 5 D. 62 E. NOTA 6 A. CD B. CB C. BA 10. In parallelogram ABCD, AB = 4 x, BC = 6 x + 5, AD = 4 x + 15. D. AC E. NOTA Find the numerical perimeter of ABCD. A. 5 B. 20 C. 90 16. In ΔABC , D is a point on AC such that ray BD D. 105 E. NOTA bisects ∠ABC. Find the length of AD when AB = 7. BC = 9, and AC = 8. 11. In ΔXYZ , XY = 8,YZ = 40, XZ = 41. A. 7 B. 9 C. 56 A. The triangle is acute. 2 2 9 B. The triangle is right. 63 D. E. NOTA C. The triangle is obtuse. 8 D. No triangle possible with these side lengths. E. NOTA 17. Given the diagram with angles as labeled and x − w = 10, find the value of y − z. 12. ΔABC is a right triangle with hypotenuse AB. D is a point on the hypotenuse such that CD ⊥ AB. If AD = 16, BD = 9 , find the length of AC. A. 4 3 B. 12 C. 15 D. 20 E. NOTA A. −10 B. 0 C. 10 13. Point X is equidistant from vertices T and N of D. 20 E. NOTA ΔTEN. Point X must lie on which of the following? A. perpendicular bisector of TN 18. The sum of the measures of the interior angles B. bisector of ∠E of a convex polygon is 2880. How many diagonals C. median to TN does this polygon have? D. the altitude to TN A. 104 B. 119 C. 135 E. NOTA D. 288 E. NOTA 14. Find the perimeter of a rhombus with diagonals 19. Given parallelogram ABCD with having lengths 24 and 32. m∠B = 60°, AD = 12, CD = 18. Find the length of the A. 40 B. 56 C. 72 altitude to AD. D. 80 E. NOTA A. 6 3 B. 9 C. 9 3 D. 12 E. NOTA Geometry Individual Test_ January Regional 20. Given the triangle with angles as marked, what 25. In rectangle ABCD, the lengths of the sides are 24 is the value of x in terms of y ? and 18. Find the perimeter of the convex quadrilateral formed by joining the midpoints of the rectangle consecutively. A. 15 B. 30 C. 60 D. 84 E. NOTA 26. ABCD is a square, and ABE is an equilateral triangle with E a point in the exterior of the square. Find the m∠AED in degrees. A. 60 B. 80 C. 100 + y 1 A. 10 ° B. 12 ° C. 15 ° D. 180 − y E. NOTA 2 D. cannot be determined E. NOTA 21. The supplement of the complement of an acute angle is always 27. ΔABC is a right isosceles triangle with right angle C. A. an acute angle B. an obtuse angle D is a point on AB and E is a point on AC such that C. a straight line D. a right angle DE || BC. If AD = 1, EC = 2 , find the value E. NOTA of AC + DB + DE. A. 3 2 B. 3 2 + 2 C. 5 + 2 22. In isosceles ΔABC x is the degree measure of D. 5 2 E. NOTA the vertex angle B and y is the degree measure of the exterior angle at B. Find the degree measure 28. In right triangle ABC with right angle at A , of ∠A in terms of x and/or y. cos C CB = 2 3, AC = 3. Find the value of. 1 sin B A. x B. 90 − y C. 180 − y 2 A. 1 B. 3 C. 2 3 1 D. 5 2 E. NOTA D. y E. NOTA 2 29. In ΔABD, AC is the altitude to BD , AC is a median, 23. If V is the measure of an exterior angle of a and ∠BAC is complementary to ∠D. Which of the regular polygon and 50° < V < 55° , how many sides following statements is false? does the polygon have? A. ∠BAC ≅ ∠DAC B. ∠BAC and ∠B are A. 5 B. 7 C. 9 complementary D. 11 E. NOTA C. ∠BAD is a right angle D. ΔABD is isosceles E. NOTA 24. In quadrilateral ABCD, diagonals AC and 30. A dog walks 2 miles north, 3 miles east, 5 miles BD are drawn. ΔADB is equilateral with north and 1 mile west. What is the shortest distance from AD = 6, BC = CD = 5. Find the length of AC. his starting position? A. 7 B. 4 + 3 2 A. 5 2 B. 11 C. 53 C. 4 + 3 3 D. 10 E. NOTA D. 65 E. NOTA Geometry Individual Test_ January Regional SOLUTIONS following to get the perimeter: 3 1 5 5+ + 6 + 20 + 30 = 62. 1. B 7. C 13. A 19. C 25. C 2 3 6 2. C 8. E 14. D 20. E 26. C 3. D 9. D 15.C 21. B 27. B 10. E Since this is a parallelogram, opposites sides 4. D 10. E 16.A 22. D 28. A are congruent making 4 x + 15 = 6 x + 5, x = 5. 5. A 11. C 17. A 23. B 29. C AB = DC = 20, BC = AD = 35 making the perimeter 6. D 12. D 18. C 24. C 30. C 110. 1. B The sum of the angles is 180 giving 11. C 412 > 402 + 82 so the triangle is obtuse. 7 x + 54 = 180, x = 18. m∠2 = 56, m∠3 = 67. 12. D The three triangles formed are similar so 57 using the geometric mean formulas, 2. C m∠1 + m∠2 = m∠3,2 x + 66 = 6 x + 9 , x =. 4 AC = AD AB = 16 25 = 20. 57 1 Substituting 2 + 3 = 31 4 2 13. A By theorem. 3. D A is true in rhombuses and squares. B is true in squares and rectangles. 14. D Diagonals are perpendicular and bisect each C is true in rhombuses and squares. other in a rhombus. So one of the triangles formed has legs of 12 and 16 making the hypotenuse which 4. D Opposites sides are congruent is a side of the rhombus 20. The perimeter is 80. therefore: y = 2 x, x + 5 = y − 5. Solve this system to get x = 10, y = 20. AB = 15. 15. C In triangle ACD, AC is longest since it is opposite a 90 degree angle. In ABC, AB is longest 5. A 180 − x = 4 (90 − x ) , x = 60 , complement is 30, since it is opposite an 86 degree angle. supplement is 120. 16. A Using the triangle angle bisector theorem, 7 9 7 6. C = , x=. x 8− x 2 7. C The lines must be in the same plane for this to 17. A Since the vertical angles are congruent, the be true. sum of the other two angles in each triangle are equal. x + y = z + w. Since we want y − z , rearrange 8. E Since each triangle is a right triangle and you always know the legs, use the Pythagorean Theorem the equation to get y − z = w − x and x − w = 10, so 4 times to get 5. z − y = − ( x − w ) , or −10. 9. D The triangles are similar by AA. First, solve 18. C First, find the number of sides of the polygon. for x. x + 7 + 6 = 2 x, x = 13. Now use proportions to ( n − 2 )180 = 2880, n = 18. Then find the number of 6 AE 3 18 15 solve for AE. = , AE = And use proportions diagonals = 135. 20 5 2. 2 7 6 1 to solve for CB, = BC = 30. Add the BC 26, 3 Geometry Individual Test_ January Regional 19. C Draw the altitude from C to AD. This forms 27. B These two triangle are similar by AA. And a 30 − 60 − 90 triangle with a hypotenuse of 18. The both are isosceles right triangles. Since altitude is opposite a 60° angle making the altitude 2 AD = 1, AE + DE =. Using proportions, 9 3. 2 2 z = x + 50, y + z + 30 = 180, z = 150 − y, AD AE 1 2 20. E = , = , BD = 2 2. 150 − y = x + 50,100 − y = x. BD EC BC 2 ⎛ 2⎞ 2 AC + DB + DE = ⎜ 2 + ⎟+2 2 + = 2+3 2 21. B The complement and the angle must both be ⎝ 2 ⎠ 2 acute making the supplement obtuse. 3 cos C 2 3 28. A = = 1or remember that the cosine 22. D Since the base angles are congruent and must sin B 3 add up to the exterior angle at the vertex angle, each 2 3 1 of one angle is the sine of the other. must be y. 2 29. C All the others can be proved true since the 23. B To solve this, look at regular polygons and the two triangles are congruent. measure of one exterior angle. # sides Exterior angle 30. C The right triangle formed by connecting the 3 120 ending point with the beginning point has legs of 7 4 90 and 2. 5 72 72 + 22 = 53. 6 60 360 3 7 = 51 7 7 24. C This is a kite by definition. So the diagonals are perpendicular and one of them is bisected. Let E be the point of intersection of the diagonals. DE = EB = 3 making AE = 3 3 and EC = 4. AC = 4 + 3 3. 25. C Use Pythagorean Theorem to find the diagonals are 30. Sides of the quadrilateral formed 1 by connecting the midpoints are the diagonals. 2 The perimeter is 60. 26. C m∠DAE = 90 + 60 = 150. Since ΔADE is 180 − 150 isosceles, ∠ADE ≅ ∠AED so each is = 15. 2 GEOMETRY INDIVIDUAL TEST January 2015 The abbreviation NOTA means "None of the Above." Diagrams 4. The complement of the supplement of may not be drawn to scale. obtuse angle A has measure x o. Which All angle measures are in degrees. is the degree measure of angle A? A. ( x − 90)o B. (180 − x)o 1. Consider the conditional statement "If a quadrilateral is a rectangle then C. ( x + 90)o D. (180 − 2 x)o it has congruent diagonals." Which is E. NOTA the converse of the contrapositive of the statement? R o 3x A. If a quadrilateral has congruent 5. diagonals then it is a rectangle. o B. If a quadrilateral is not a rectangle then (x+94) 32 o it does not have congruent diagonals. S T U C. If a quadrilateral does not have congruent diagonals then it is not a S, T and U are collinear, and in ∆TUR , rectangle. angles R, U and exterior angle ∠RTS D. If a quadrilateral is a rectangle then it has congruent diagonals. have measures given in the diagram. E. NOTA Find the measure of ∠RTS in degrees. A. 125 o B. 124 o D C C. 115 o D. 112.5 o 2. m∠ABC = 4 x + 4 and E. NOTA m∠CBD = 50 − x. If BD ⊥ BA then find m∠ABC B A in degrees. 6. ∆RST ~ ∆DEF. RS=12, ST=8, DE=5, A. 9.2 o B. 12 o m∠R = 2 x + 10 and m∠D = 40. Give the value of x. C. 41.8 o D. 52 o E. NOTA A. 53 B. 43 C. 25 D. 15 E. NOTA 3. A B C 7. A right triangle has one side with length 12 and all sides are integers. Find the sum of the lengths of all of the possible D E F hypotenuses of the triangles. A. 28 B. 48 C. 85 D. 97 AC is parallel to DF , with transversal BE. E. NOTA A, B and C are collinear, as are D, E and F. m∠ABE = 3 x + 5 and m∠DEB = 2 x + 15. Find m∠CBE in degrees. A. 10 o B. 32 o o C. 35 D. 79 o E. NOTA GEOMETRY INDIVIDUAL TEST January 2015 8. ∆RST is isosceles with two of its angles measuring ( x + 10)o and (2 x − 10)o. P G Which could NOT be a degree measure 12. of one angle of ∆RST ? A. 20 o B. 30 o C.48 o D. 75 o U T E. NOTA Right ∆PUT has altitude to the hypotenuse UG. If PG= x + 2 and GT= x + 7 and GU= 3 x for x > 0 , then 9. AB bisects ∠RAT. If m∠BAT = x + 30 find the length GU. and m∠RAT = 3 x − 10 then find m∠BAR 7 in degrees. A. B. 2 8 A. 100 o B. 80 o C. 3 D. 6 C. 70 o D. 60 o E. NOTA E. NOTA 10. d 13. 3 8 P A flag pole is 15 feet in height and 4 P 6 is x feet horizontally from a point P Two vertical poles are 4 feet and 6 feet on the ground. Angle P is the angle of from point P as shown. The heights of the elevation from P to the top of the flag poles are 3 and 8 feet as shown. Find the pole and sin P = 0.6. Give the value distance between the tops of the poles (d) of x in feet. above, in feet. A. 18 B. 20 C. 22 D. 24 A. 5 5 B. 5 3 E. NOTA C. 5 2 D. 12 E. NOTA 11. Which three numbers could be the lengths of the sides of an obtuse 14. In ∆RST , RS=5 and ST=6. In ∆ABC , triangle? AB=5 and BC=6. Which could NOT A. 3, 4, 5 B. 6, 8, 9.5 necessarily be used to prove that C. 7, 24, 26 D. 9, 40, 40.5 ∆RST ≅ ∆ABC ? E. NOTA A. ∠RST ≅ ∠ABC B. AC ≅ RT C. ∠R, ∠A are right angles D. ∠RST , ∠ABC are right angles E. NOTA GEOMETRY INDIVIDUAL TEST January 2015 T 15. U 18. 8 6 B
