Regents Geometry Exam — Questions and Answers

Question 1: Which transformation preserves both size and shape but changes orientation?

Shear
Reflection (Correct answer)
Stretch
Dilation

Correct answer: Reflection

A reflection is an isometry that reverses orientation while keeping size and shape.

Question 2: In similar triangles, the ratio of corresponding medians equals the ratio of:

Corresponding angles
Corresponding areas
The perimeters squared
Corresponding sides (Correct answer)

Correct answer: Corresponding sides

All corresponding linear measures, including medians, share the side ratio.

Question 3: What is a diameter?

A line outside the circle.
A line inside the circle not touching the center.
A line through the center connecting two points on the circle (Correct answer)
A curve inside the circle.

Correct answer: A line through the center connecting two points on the circle

A diameter is a straight line segment that passes through the exact center of a circle and connects two points on its circumference. It represents the longest possible chord in a circle. The diameter is a crucial measurement, as it is twice the length of the radius.

Question 4: Which property of equality is used to justify 'If x = 5, then x + 3 = 8'?

Reflexive property
Symmetric property
Addition property of equality (Correct answer)
Subtraction property of equality

Correct answer: Addition property of equality

Adding the same number to both sides is the addition property of equality.

Question 5: The statement 'a = a' is an example of which property of equality?

Symmetric property
Substitution property
Reflexive property (Correct answer)
Transitive property

Correct answer: Reflexive property

The reflexive property of equality states that any quantity equals itself.

Question 6: Two lines that intersect to form right angles are said to be:

Congruent
Skew
Parallel
Perpendicular (Correct answer)

Correct answer: Perpendicular

Perpendicular lines meet at 90-degree angles.

Question 7: What is the total surface area of a cylinder with radius 2 and height 3? (use π ≈ 3.14)

25.12
50.24
37.68
62.8 (Correct answer)

Correct answer: 62.8

SA = 2πr² + 2πrh = 2(3.14)(4) + 2(3.14)(2)(3) = 25.12 + 37.68 = 62.8.

Question 8: The Law of Cosines c² = a² + b² − 2ab cos C reduces to the Pythagorean theorem when C equals what?

0°
90° (Correct answer)
45°
60°

Correct answer: 90°

At C = 90°, cos C = 0, leaving c² = a² + b².

Question 9: What is volume?

The weight of an object.
The space inside a 2D shape.
The space inside a 3D object (Correct answer)
The length around a 2D shape.

Correct answer: The space inside a 3D object

Volume is a measure of the three-dimensional space occupied by a solid object or enclosed within a container. Unlike area, which is for 2D shapes, volume quantifies the capacity or extent of a 3D object. It is typically measured in cubic units.

Question 10: A circle has area 49π. What is its circumference?

28π
7π
14π (Correct answer)
49π

Correct answer: 14π

Area 49π gives r = 7, so circumference = 2π(7) = 14π.

Question 11: Two similar triangles have areas in the ratio 25:49. Their perimeters are in the ratio:

25:49
625:2401
12.5:24.5
5:7 (Correct answer)

Correct answer: 5:7

Perimeter ratio is the square root of the area ratio: √25:√49 = 5:7.

Question 12: An angle measuring exactly 180° is called what?

Obtuse
Right
Straight (Correct answer)
Reflex

Correct answer: Straight

A straight angle measures exactly 180°.

Question 13: Which similarity criterion uses two pairs of proportional sides and the included angle?

AA similarity
SSS similarity
HL similarity
SAS similarity (Correct answer)

Correct answer: SAS similarity

SAS similarity requires two proportional sides with equal included angles.

Question 14: Which property distinguishes a square from a rhombus?

Four right angles (Correct answer)
Diagonals bisect each other
Four equal sides
Perpendicular diagonals

Correct answer: Four right angles

A square must have four right angles, which a general rhombus lacks.

Question 15: The sum of any two sides of a triangle must be:

Greater than the third side (Correct answer)
Equal to the third side
Equal to 180 degrees
Less than the third side

Correct answer: Greater than the third side

The triangle inequality requires any two sides to sum to more than the third.

Question 16: How many edges does a triangular prism have?

9 (Correct answer)
6
7
8

Correct answer: 9

A triangular prism has 9 edges: 3 on each triangular base and 3 lateral edges connecting them.

Question 17: A dilation with scale factor 3 centered at the origin maps (2, -1) to?

(2/3, -1/3)
(5, 2)
(6, 3)
(6, -3) (Correct answer)

Correct answer: (6, -3)

Multiply each coordinate by 3: (6, -3).

Question 18: What is a chord in a circle?

The outer edge of the circle.
A line that touches the circle at one point.
A segment connecting two points on the circle (Correct answer)
A line outside the circle.

Correct answer: A segment connecting two points on the circle

A chord in a circle is defined as any straight line segment that connects two distinct points on the circle's circumference. It does not necessarily pass through the center. The longest possible chord in a circle is its diameter.

Question 19: In which quadrant is sine positive but cosine negative?

Second (Correct answer)
Fourth
Third
First

Correct answer: Second

In quadrant II, sine is positive and cosine is negative.

Question 20: What is a dilation in geometry?

A turn.
A slide.
A flip.
A proportional resizing (Correct answer)

Correct answer: A proportional resizing

A dilation is a geometric transformation that changes the size of a figure by either enlarging or reducing it, while maintaining its shape. This resizing is proportional, meaning all dimensions are scaled by the same factor, called the scale factor, from a fixed center point. Unlike other transformations, it does not preserve size.

Question 21: A rectangular garden is 15 m by 8 m. How much fencing is needed to enclose it?

46 m (Correct answer)
60 m
120 m
23 m

Correct answer: 46 m

Fencing equals the perimeter: 2(15 + 8) = 46 m.

Question 22: What is the slope of a line?

The length of the line.
The y-intercept.
The midpoint.
The ratio of vertical to horizontal change (Correct answer)

Correct answer: The ratio of vertical to horizontal change

The slope of a line is a measure of its steepness and direction. It is defined as the ratio of the vertical change (rise) to the horizontal change (run) between any two distinct points on the line. A positive slope indicates an upward trend, while a negative slope indicates a downward trend.

Question 23: Equal chords in the same circle are always equidistant from the:

Center (Correct answer)
Nearest tangent
Longest chord
Circumference

Correct answer: Center

Congruent chords are the same distance from the circle's center.

Question 24: An angle measures exactly 90°. What type of angle is it?

Obtuse angle
Right angle (Correct answer)
Acute angle
Straight angle

Correct answer: Right angle

An angle of exactly 90° is a right angle.

Question 25: The arc opposite a 40° inscribed angle measures:

80° (Correct answer)
40°
160°
20°

Correct answer: 80°

The intercepted arc is twice the inscribed angle: 2 × 40° = 80°.

Question 26: A composite shape is made of a 4 cm by 3 cm rectangle joined to a 4 cm by 2 cm rectangle. What is its total area?

12 cm²
24 cm²
14 cm²
20 cm² (Correct answer)

Correct answer: 20 cm²

Add the two areas: (4 × 3) + (4 × 2) = 12 + 8 = 20 cm².

Question 27: Triangle ABC ~ triangle DEF. If AB = 8, DE = 12, and BC = 10, what is EF?

6.7
15 (Correct answer)
13
14

Correct answer: 15

The ratio is 8/12 = 10/EF, so EF = 15.

Question 28: How do you construct the perpendicular bisector of a line segment?

Draw a line parallel to the segment
Measure the segment and mark the midpoint, then draw a line through it
Use a protractor to measure 90° at the midpoint
Draw arcs from each endpoint with equal radii greater than half the segment's length, then connect the intersection points (Correct answer)

Correct answer: Draw arcs from each endpoint with equal radii greater than half the segment's length, then connect the intersection points

Open the compass to more than half the segment's length, draw intersecting arcs from both endpoints, and connect the two intersection points.

Question 29: What is the area of a circle with a radius of 5 cm? (Use π ≈ 3.14)

78.5 cm² (Correct answer)
25 cm²
31.4 cm²
15.7 cm²

Correct answer: 78.5 cm²

Area is πr² = 3.14 × 25 = 78.5 cm².

Question 30: The diagonals of which quadrilateral are always equal in length AND bisect each other?

Trapezoid
Rectangle (Correct answer)
Rhombus
Kite

Correct answer: Rectangle

A rectangle's diagonals are congruent and bisect each other.

Question 31: Using the Law of Sines, if a = 8, angle A = 30°, and angle B = 45°, which sets up side b?

b = 8 sin 30° sin 45°
b = 8 / (sin 45° sin 30°)
b = 8 sin 45° / sin 30° (Correct answer)
b = 8 sin 30° / sin 45°

Correct answer: b = 8 sin 45° / sin 30°

By the Law of Sines a/sin A = b/sin B, so b = 8 sin 45°/sin 30°.

Question 32: A trapezoid has parallel sides of 6 cm and 10 cm and a height of 4 cm. What is its area?

24 cm²
16 cm²
40 cm²
32 cm² (Correct answer)

Correct answer: 32 cm²

Area = ½(b₁+b₂)h = ½(6+10)(4) = 32 cm².

Question 33: What is the locus of all points equidistant from two parallel lines?

A circle centered between them
A third line parallel to and midway between them (Correct answer)
The perpendicular bisector of both lines
Two intersecting lines

Correct answer: A third line parallel to and midway between them

The locus of points equidistant from two parallel lines is a single line parallel to both and midway between them.

Question 34: If leg segments on the hypotenuse are 4 and 16, the altitude to the hypotenuse is:

8 (Correct answer)
64
20
10

Correct answer: 8

The altitude is the geometric mean: √(4 × 16) = √64 = 8.

Question 35: What does sin(2θ) equal by the double-angle formula?

sin²θ − cos²θ
2 sin θ cos θ (Correct answer)
1 − 2 sin²θ
2 cos²θ − 1

Correct answer: 2 sin θ cos θ

The double-angle formula gives sin 2θ = 2 sin θ cos θ.

Question 36: What is true about corresponding angles in similar triangles?

They are doubled.
They are proportional.
They are halved.
They are equal (Correct answer)

Correct answer: They are equal

A defining characteristic of similar triangles is that their corresponding angles are congruent, meaning they are equal in measure. While the side lengths of similar triangles are proportional, the angles themselves do not change. This property is crucial for identifying and working with similar figures.

Regents Geometry Exam

The Regents Geometry Exam assesses student mastery of high school geometry concepts including congruence, similarity, right triangles, trigonometry, circles, coordinate geometry, and geometric modeling, aligned to Common Core standards.

Questions

180 min

Time Limit

65%

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