Geometry and Volume of Shapes Quiz

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Questions and Answers

What is the volume of a cuboid?

  • $2(lb + bh + hl)$
  • $lbh$ (correct)
  • $l + b + h^2$
  • $l + b + h$

A cube has a total surface area of $6a^2$.

True (A)

What is the length of the body diagonal of a cuboid?

$ oot{l^2 + b^2 + h^2}$

The radius of the circumscribed sphere around a cuboid is _____

<p>$\frac{1}{2}\sqrt{l^2 + b^2 + h^2}$</p> Signup and view all the answers

Match the following geometric shapes with their properties:

<p>Cuboid = Rectangular faces Cube = All faces are squares Prism = Ends are congruent figures Sphere = All points equidistant from center</p> Signup and view all the answers

What condition must be met for a sphere to be inscribed in a cuboid?

<p>All dimensions must be equal (B)</p> Signup and view all the answers

A prism is defined by having two ends that are not congruent figures.

<p>False (B)</p> Signup and view all the answers

What is Euler's formula relating the number of faces, vertices, and edges of a solid?

<p>F + V = E + 2</p> Signup and view all the answers

What is the length of AD in triangle ABC, where AB = 7 and AC = 25?

<p>13.0 (D)</p> Signup and view all the answers

The area of triangle PQR is less than the area of triangle ABC if the area of triangle ABC is 10 square centimeters.

<p>False (B)</p> Signup and view all the answers

How deep is the lake if a water lily with a stem extends one meter above the surface and bends under water three meters from the original point?

<p>6</p> Signup and view all the answers

In triangle ABC, if both incircles of triangles ABD and ACD touch AD at a common point E, the length of CD is _____

<p>8</p> Signup and view all the answers

In an isosceles right triangle with a square inscribed and where the ratio of x to y is 2:1, what is the ratio of the area of the square to the area of the triangle?

<p>2:3 (B)</p> Signup and view all the answers

Match the geometric terms with their definitions:

<p>Angle Bisector = A segment that divides an angle into two equal angles. Equilateral Triangle = A triangle with all sides of equal length. Isosceles Triangle = A triangle with at least two equal sides. Area = The measure of the space inside a shape.</p> Signup and view all the answers

If AC = AP, BC = CR, and AB = BQ in triangle ABC, the area of triangle ABC cannot be determined.

<p>False (B)</p> Signup and view all the answers

What is the value of angle DEC in degrees, given that ABCD is a square and ABE is an equilateral triangle?

<p>30</p> Signup and view all the answers

What happens to the radius of a circle when a straight line is tangent to it?

<p>The radius is perpendicular to the tangent line. (A)</p> Signup and view all the answers

The lengths of two tangent segments drawn from an exterior point to a circle are always unequal.

<p>False (B)</p> Signup and view all the answers

If the length of tangent segment AP is 11, what is the perimeter of triangle ABC?

<p>22</p> Signup and view all the answers

The angle that a tangent makes with a chord at the point of contact is equal to the angle subtended by the chord in the ______.

<p>alternate segment</p> Signup and view all the answers

Match the descriptions with the corresponding properties:

<p>Tangent length from exterior point = The lengths are equal. Angle made by tangent and chord = Equal to angle subtended in alternate segment. Radius to tangent point = Is perpendicular to tangent. Line connecting external point to center = Bisects the angle between the tangents.</p> Signup and view all the answers

From an external point P, if PC = 6 and PA = 5, what is the relationship between PA, PB, and PC?

<p>PA × PB = PC² (B)</p> Signup and view all the answers

The line joining the exterior point to the center does not bisect the angle between the tangents.

<p>False (B)</p> Signup and view all the answers

If PA = 18 and PB = 12, what would be PE if PB × PA = PE × PD given PD = 2?

<p>16</p> Signup and view all the answers

What is the length of AB in triangle ABC?

<p>26 + 2√38 (B)</p> Signup and view all the answers

The ratio AD:DC in triangle ABC is 3:1.

<p>False (B)</p> Signup and view all the answers

What is the sum of the angles at the five points of a star formed by points A, B, C, D, and E?

<p>540°</p> Signup and view all the answers

The diameter of the inscribed circle in a trapezoid with parallel sides of 75 and 108 units is ____.

<p>90</p> Signup and view all the answers

What is the ratio of the area of the circumscribing circle of a square to the area of the inscribed circle of an equilateral triangle?

<p>16:9 (B)</p> Signup and view all the answers

It is possible for two circles to touch each other externally and also touch a bigger circle internally.

<p>True (A)</p> Signup and view all the answers

If a cubic container with an edge of 16 cm is 5/8 full of liquid, what is the length of line segment LC when tilted?

<p>9</p> Signup and view all the answers

Match the following expressions for the radius of the circle inscribing hexagon ABCDEF.

<p>(x^2 + y^2 + xy)/3 = Option A (x^2 + y^2 + xy)/2 = Option B (x^2 + y^2 - xy)/3 = Option C (x^2 + y^2 - xy)/2 = Option D</p> Signup and view all the answers

If angle D in triangle DEF is 40 degrees, what is angle ACB?

<p>70 (B)</p> Signup and view all the answers

A triangle with sides a, b, and c satisfies the condition a + b + c = bc + ca + ab is always an equilateral triangle.

<p>True (A)</p> Signup and view all the answers

What is the perimeter of triangle PQR given that it consists of circles with radius 20 and lengths AB=5, CD=10, EF=12?

<p>66</p> Signup and view all the answers

If AD=24 and BC=12, the ratio of the area of triangle CBE to that of triangle ADE is ___

<p>1:4</p> Signup and view all the answers

Match the geometric figures with their corresponding properties:

<p>Equilateral triangle = All sides are equal Rectangle = Opposite sides are equal Right triangle = One angle is 90 degrees Circle = All points equidistant from the center</p> Signup and view all the answers

In triangle ABC, angle B is a right angle and side AC = 6 cm. If D is the midpoint of AC, what is the length of BD?

<p>3 cm (D)</p> Signup and view all the answers

If AB is perpendicular to BC, and CE bisects angle C with ∠A = 30°, then ∠CED is 60 degrees.

<p>False (B)</p> Signup and view all the answers

What is the area of triangle ABC if the chord CA is 5 cm long and the radius is 6.5 cm?

<p>calculation required</p> Signup and view all the answers

In triangle PQR, if ∠PQR is 30° and PQ and PR are the angle bisectors of angles APB and APC respectively, what are the measurements of the other two angles in triangle PQR?

<p>70° and 80° (B)</p> Signup and view all the answers

In triangle PQR, PS bisects ∠QPR. The area of triangle PQS is 40 sq. cm, and the area of triangle PQR is 70 sq. cm.

<p>False (B)</p> Signup and view all the answers

What is the length of MR in triangle MNO, given that MO = 30 cm and NQ bisects MP?

<p>10 cm</p> Signup and view all the answers

The maximum possible length of QS, given that side PR is 8cm and both QR and SR take integral values greater than 1, is _____ cm.

<p>32</p> Signup and view all the answers

In a right-angled triangle with sides p, q, r where $p < q < r$, if $2p + 7r = 9q$ and $p = 12cm$, what is the value of r?

<p>26.5 cm (D)</p> Signup and view all the answers

Match the elements with their corresponding properties:

<p>PS = Angle bisector PD = Median MO = Side length NQ = Angle bisector</p> Signup and view all the answers

In triangle PQR, if the point D is the midpoint of side QR, and FQ = _____ cm, then the length of side PQ is determined based on the properties of midpoints.

<p>3</p> Signup and view all the answers

How do you determine the value of PO in triangle MNO when MN = 9cm, MO = 12cm, and PO = PN + 1?

<p>5 cm</p> Signup and view all the answers

Flashcards

Cuboid

A parallelepiped with all rectangular faces. It has three dimensions: length (l), breadth (b), and height (h).

Body Diagonal of a Cuboid

The line segment connecting opposite vertices of a cuboid. There are four body diagonals, all equal in length.

Face Diagonal of a Cuboid

The distance along a face of a cuboid, connecting opposite vertices. There are three pairs of face diagonals.

Prism

A solid with two parallel, congruent ends, and side faces that are parallelograms. The ends can be triangles, quadrilaterals, or polygons.

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Right Prism

A prism where the side edges are perpendicular to the ends. This results in rectangular side faces.

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Cube

A special case of a cuboid where all faces are squares. It has only one dimension: side length (a).

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Body Diagonal of a Cube

The line segment connecting opposite vertices of a cube. There are four body diagonals, all equal in length.

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Face Diagonal of a Cube

The line segment connecting opposite vertices of a cube's face. All face diagonals are equal in length.

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Angle Bisectors and Triangle Angles

The angle bisectors of two angles of a triangle divide the third angle into two angles which are equal to half the sum of the first two angles.

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Area of Triangle

The area of a triangle is directly proportional to the base and height. If the base is increased by a factor of 2.5, the area will also increase by the same factor.

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Median of a Triangle

The median of a triangle divides the triangle into two triangles with equal areas. The line segment connecting the midpoint of a side to the opposite vertex is called a median.

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Similar Triangles

In a triangle, the ratio of the lengths of the corresponding sides of similar triangles is equal. If two angles of one triangle are equal to two angles of another triangle, then the two triangles are said to be similar.

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Pythagorean Theorem

The Pythagorean theorem states that in a right-angled triangle, the square of the hypotenuse is equal to the sum of the squares of the other two sides. The hypotenuse is the longest side of a right-angled triangle.

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Sliding a Triangle

When a triangle is moved along a line, the area of the overlapping region is equal to half the area of the original triangle.

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Parallel lines and Transversal

Parallel lines cut by a transversal create corresponding angles. They are equal in measure. The ratio of lengths of corresponding line segments is the same.

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Angle Bisector Theorem

The angle bisector theorem states that the bisector of an angle of a triangle divides the opposite side into segments that are proportional to the other two sides of the triangle. The angle bisector is a line segment that divides an angle into two equal angles.

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Equilateral Triangle Property

In a triangle, if the sum of the lengths of all the sides is equal to the sum of the squares of the lengths of all the sides, then the triangle is an equilateral triangle.

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Perimeter of a Triangle

The perimeter of a triangle is the total length of all its sides. In the given case, we need to find the sum of the lengths of the sides of triangle PQR.

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Midpoint Theorem

The line segment joining the midpoints of two sides of a triangle is parallel to the third side and half its length.

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Congruent Triangles

Angles at the same position in two congruent triangles are equal, as well as corresponding sides being equal. This allows us to find missing angles and sides in similar triangles.

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Angle-Side Relationship

A perpendicular line drawn from the vertex of a triangle to its opposite side divides the triangle into two right-angled triangles. This helps us find relationships between angles and sides in a triangle.

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Angle Subtended at Centre & Circumference

The measure of the angle at the centre of a circle is twice the measure of the angle at the circumference when both angles are subtended by the same arc.

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What is the ratio AD:DC?

The ratio of the length of the line segment AD to the length of the line segment DC is 3:2.

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What is the sum of the angles at the five points?

The sum of the angles at the five points of a star formed by connecting five points on a circle is 540 degrees.

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What is the diameter of the inscribed circle?

The diameter of the inscribed circle of an isosceles trapezoid with parallel sides of lengths 75 and 108 units is 91.5.

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What is the ratio of the area of the circle circumscribing the square to that of the circle inscribed in the triangle?

The ratio of the area of the circle circumscribing the square to the area of the circle inscribed in the equilateral triangle is 27:8.

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What is the length of line segment LC?

The length of line segment LC is 12 cm. Given the cube is 5/8th full with liquid, calculate the height of the liquid and use ratios to find the length of LC.

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What is the radius of the circle?

The radius of the circle is [(x² + y² - xy)/3]1/2 units.

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What is the value of the ratio AD: DC?

The value of the ratio AD:DC is 7:3.

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What is the length of AB?

The length of AB is 26 + 2√38 units.

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Tangent to a circle

A line that touches a circle at exactly one point.

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Radius & Tangent Relation

The radius drawn to the point of contact of a tangent is perpendicular to the tangent.

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Equal Tangent Segments

If two tangents are drawn to a circle from an external point, the lengths of the tangent segments are equal.

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Secant-Tangent Theorem

If a secant and a tangent are drawn from an external point to a circle, the product of the lengths of the secant segment and its external segment is equal to the square of the tangent segment.

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Tangent-Chord Theorem

The angle between a tangent and a chord drawn from the point of contact is equal to the angle subtended by that chord in the alternate segment of the circle.

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Finding Length of Angle Bisector

In a right-angled triangle with sides AB = 7, AC = 25, and AD as the angle bisector of angle A, AD is also a perpendicular bisector of BC. Therefore, AD divides the triangle ABC into two smaller congruent triangles (ABD and ACD). The perpendicular bisector divides the hypotenuse into two segments, with the segments in the same ratio as the legs of the original right triangle. Therefore, using the Pythagorean theorem to find BC, we get BC = 24. Since D bisects BC, BD = 12. Applying the Pythagorean theorem to triangle ABD, we get AD = 13.

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Tangent-Tangent Property and Incircles

The incircle of a triangle is a circle that is tangent to all three sides of the triangle. The incenter of a triangle is the center of the incircle and it is the intersection point of the angle bisectors. It is known that the length of the tangent from a point to a circle is equal from all points on the circle. If the incircles of triangles ABD and ACD touch AD at a common point E, then the tangents from D to the incircles of both triangles are equal. Therefore, applying the tangent-tangent property, we can conclude that BD = CD.

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Finding Ratio of Areas in Inscribed Shapes

In an isosceles triangle ABC with a right angle at B, a square is inscribed with three vertices on the sides of the triangle. The ratio of the side of the square to the base of the isosceles triangle is 2:1. Finding the area of the square and the triangle, we get the ratio of their areas as 2:5.

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Finding Area of Expanded Triangle

In triangle ABC, sides AB, AC, and BC are extended till Q, P and R such that AC = AP, BC = CR, and AB = BQ. This creates three new isosceles triangles: ACP, BCR, and ABQ. Since the area of triangle ABC is 10 sq cm, and each new triangle has double the base and height, the area of each new triangle is 40 sq cm. The area of Triangle PQR is simply the sum of the areas of these new triangles plus the area of the original triangle. Therefore, the area of triangle PQR is 40 + 40 + 40 + 10 = 130 sq cm.

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Applying Pythagorean Theorem to Water Lily Problem

When the water lily bends at the bottom of its stem, it creates a right triangle with the stem as the hypotenuse, the distance from the point where it originally came out of the water to the point where it disappears as one leg, and the depth of the lake as the other leg. Using the Pythagorean theorem, we can solve for the depth of the lake, which is 5m.

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Finding Angle Measurement in Geometry

In the figure, ABCD is a square and ABE is an equilateral triangle. The measure of each angle in an equilateral triangle is 60 degrees. Since AB is a common side to both the square and the equilateral triangle, angle ABE is 60 degrees. Angle ABC is 90 degrees as it is an angle of the square. Therefore, angle CBE is 90 - 60 = 30 degrees. Similarly, angle DEC is also 30 degrees.

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Angle Bisector Property

In triangle ABC, D and E are any points on AB and AC such that AD = AE. This means triangle ADE is an isosceles triangle. Since AD = AE, angles AED and ADE are equal. The sum of angles in a triangle is 180 degrees. Therefore, angle DAE = 180 - 2 * angle AED. Therefore, angle DAE is twice the measure of angle AED, which means that AD is the angle bisector of angle A.

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Height of Crossing Wires

If two wires are completely taut and cross each other, they form a right angle at the point of intersection. The height at which they cross is the perpendicular distance from the point of intersection to the ground. In this case, the two wires are represented as 'a' and 'b'. Because of similar triangles, the height at which the wires cross is equal to

2ab
-----
a + b
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Study Notes

Geometry: Lines and Angles

  • A point is an exact location, represented by a dot. It has no magnitude.
  • A line segment is a straight path between two points, possessing a definite length.
  • A ray is a line segment that extends infinitely in one direction.
  • Intersecting lines are lines that share a common point called the point of intersection.
  • Concurrent lines are two or more lines that intersect at the same point.
  • An angle is formed when two straight lines meet at a point. The common point is called the vertex.
  • A right angle measures 90°.
  • An acute angle measures less than 90°.
  • An obtuse angle measures more than 90° but less than 180°.
  • A reflex angle measures more than 180° but less than 360°.
  • Complementary angles add up to 90°.
  • Supplementary angles add up to 180°.
  • Vertically opposite angles are equal.

Geometry: Triangles

  • The sum of the three angles in a triangle always equals 180°.
  • The sum of any two sides of a triangle is greater than the third side.
  • In a right-angled triangle, the square of the hypotenuse is equal to the sum of the squares of the other two sides (Pythagorean theorem).
  • Similar triangles have the same shape but may differ in size. Corresponding angles are equal, and corresponding sides are in proportion.
  • The ratio of the areas of similar triangles is equal to the square of the ratio of their corresponding sides.
  • The medians of a triangle intersect at a point called the centroid, which divides each median in the ratio 2:1.
  • The altitude of a triangle is a line segment from a vertex perpendicular to the opposite side.
  • The angle bisectors of a triangle intersect at a point called the incenter.

Geometry: Polygons

  • A polygon is a closed two-dimensional shape formed by straight lines.
  • The sum of the interior angles of a polygon with n sides is given by (n-2) × 180°.
  • A regular polygon has all sides and angles equal.
  • A quadrilateral is a polygon with four sides. Common quadrilaterals include squares, rectangles, parallelograms, and rhombuses.
  • A trapezoid is a quadrilateral with at least two parallel sides.

Geometry: Circles

  • A circle is a set of points equidistant from a fixed point called the center.
  • The radius of a circle is the distance from the center to any point on the circle.
  • The diameter of a circle is twice the radius.
  • The circumference of a circle is the distance around the circle.
  • The area of a circle is given by the formula Ï€r².
  • A chord is a line segment connecting two points on the circle.
  • A tangent is a line that touches the circle at only one point.
  • The angle subtended by an arc of a circle at the center is twice the angle subtended by the same arc at any point on the remaining part of the circumference.
  • Equal chords in a circle are equidistant from the center.

Geometry: Solids

  • A solid is a three-dimensional shape bounded by surfaces.
  • The volume of a solid is the amount of space it occupies.
  • A cube is a three-dimensional shape with six square faces that are congruent. The length of the body diagonal is √3 a, and the surface area= 6a². The volume= a³
  • A cuboid is a three-dimensional shape with six rectangular faces.
  • A pyramid has a polygon base and triangular faces meeting at a common point called the apex.
  • A cone has a circular base and a curved surface that meets at a point called the vertex.
  • A sphere is a set of points in space equidistant from a fixed point (the center).
  • The surface area of a sphere is 4Ï€r². The volume of a sphere is (4/3)Ï€r³.

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