January Regional Geometry Individual Test PDF

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)...

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