Regents Geometry Exam — Questions and Answers

Question 1: Which quadrilateral has diagonals that are perpendicular but NOT necessarily equal?

Rhombus (Correct answer)
Rectangle
Square
Isosceles trapezoid

Correct answer: Rhombus

A rhombus has perpendicular diagonals that are generally different lengths.

Question 2: The ambiguous case (two possible triangles) can occur with which given information?

ASA
SSS
SSA (Correct answer)
SAS

Correct answer: SSA

SSA can produce zero, one, or two triangles, making it ambiguous.

Question 3: A trapezoid has parallel sides of 6 cm and 10 cm and a height of 4 cm. What is its area?

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

Correct answer: 32 cm²

Trapezoid area is ½(a + b) × h = ½(6 + 10) × 4 = 32 cm².

Question 4: The identity 1 + tan²θ equals:

csc²θ
sec²θ (Correct answer)
cot²θ
sin²θ

Correct answer: sec²θ

Dividing sin²θ + cos²θ = 1 by cos²θ gives 1 + tan²θ = sec²θ.

Question 5: A central angle measures 72°. What fraction of the full circle does its arc represent?

2/5
1/3
1/4
1/5 (Correct answer)

Correct answer: 1/5

72° ÷ 360° = 1/5 of the circle.

Question 6: What is the midpoint of the segment joining (-2, 3) and (4, 7)?

(3, 2)
(1, 4)
(1, 5) (Correct answer)
(2, 10)

Correct answer: (1, 5)

Midpoint = ((-2+4)/2, (3+7)/2) = (1, 5).

Question 7: In a two-column proof, the left column contains statements and the right column contains:

Conclusions only
Diagrams
Given information only
Reasons or justifications (Correct answer)

Correct answer: Reasons or justifications

Each statement in a two-column proof must be justified by a reason on the right.

Question 8: The Hypotenuse-Leg (HL) theorem applies only to which type of triangle?

Equilateral triangles
Obtuse triangles
Right triangles (Correct answer)
Scalene triangles

Correct answer: Right triangles

HL congruence is valid only for right triangles because it references the hypotenuse.

Question 9: What is the measure of an angle that is supplementary to a 118° angle?

62° (Correct answer)
42°
72°
82°

Correct answer: 62°

Supplementary angles sum to 180°, so 180 − 118 = 62°.

Question 10: What is true about corresponding angles in similar triangles?

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

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.

Question 11: What is the formula for the perimeter of a rectangle?

Length ÷ width.
2 × (length + width) (Correct answer)
Length + width.
Length × width.

Correct answer: 2 × (length + width)

The perimeter of a rectangle is the total distance around its boundary. A rectangle has two lengths and two widths. Therefore, to find the perimeter, you add the length and width together and then multiply the sum by two to account for all four sides.

Question 12: In an isosceles trapezoid, the base angles are:

Congruent (Correct answer)
Right angles
Supplementary to each other
Complementary

Correct answer: Congruent

An isosceles trapezoid has congruent base angles along each parallel side.

Question 13: What is the area of a triangle with base 10 and height 6?

60
120
30 (Correct answer)
16

Correct answer: 30

Area = ½·base·height = ½(10)(6) = 30.

Question 14: What is the distance formula used for?

To find the slope.
To find the midpoint.
To calculate distance between two points (Correct answer)
To find the area.

Correct answer: To calculate distance between two points

The distance formula is derived from the Pythagorean theorem and is used to calculate the straight-line distance between any two points in a coordinate plane. Given two points (x1, y1) and (x2, y2), it finds the length of the line segment connecting them.

Question 15: How do you construct a perpendicular to a line through a point ON the line?

Measure 90° with a protractor at the point
Draw two arcs from the point along the line, then bisect the resulting segment above the line (Correct answer)
Draw a circle at the point and find where it crosses the line
Copy the angle adjacent to the point

Correct answer: Draw two arcs from the point along the line, then bisect the resulting segment above the line

Place the compass at the point, mark two equidistant points on the line, then construct the perpendicular bisector of that segment.

Question 16: A transversal creates a 55° angle with line l. If l ∥ m, what is the co-interior angle on the same side with line m?

55°
125° (Correct answer)
135°
115°

Correct answer: 125°

Co-interior angles are supplementary, so the angle with line m is 180° − 55° = 125°.

Question 17: Which transformation preserves both size and shape but changes orientation?

Shear
Stretch
Reflection (Correct answer)
Dilation

Correct answer: Reflection

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

Question 18: The Law of Cosines c² = a² + b² − 2ab cos C reduces to the Pythagorean theorem when C equals what?

90° (Correct answer)
0°
60°
45°

Correct answer: 90°

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

Question 19: Which similarity criterion uses two pairs of proportional sides and the included angle?

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

Correct answer: SAS similarity

SAS similarity requires two proportional sides with equal included angles.

Question 20: What is a geometric proof?

A logical argument showing why a statement is true (Correct answer)
A random drawing.
A guess about geometry.
A set of measurements.

Correct answer: A logical argument showing why a statement is true

A geometric proof is a systematic and logical argument that uses definitions, postulates, and previously proven theorems to demonstrate the truth of a geometric statement. It involves a series of steps, each justified by a known fact or rule, leading to a conclusive statement.

Question 21: What is an example of a geometric theorem?

Triangle postulate.
Parallel postulate.
Angle addition postulate.
Pythagorean theorem (Correct answer)

Correct answer: Pythagorean theorem

The Pythagorean theorem is a fundamental geometric theorem that states that in a right-angled triangle, the square of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the other two sides. Unlike postulates, theorems are statements that can be proven using definitions, postulates, and previously established theorems.

Question 22: The opposite angles of a parallelogram are:

Supplementary
Complementary
Congruent (Correct answer)
Right angles

Correct answer: Congruent

Opposite angles in a parallelogram are always equal in measure.

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

Circumference
Center (Correct answer)
Nearest tangent
Longest chord

Correct answer: Center

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

Question 24: If DE is parallel to BC in triangle ABC with AD = 4, DB = 6, and AE = 6, what is EC?

4
10
9 (Correct answer)
6

Correct answer: 9

By proportionality 4/6 = 6/EC, so EC = 9.

Question 25: What is the y-intercept of the line 2x + 3y = 12?

12
4 (Correct answer)
6
-4

Correct answer: 4

Set x = 0: 3y = 12, so y = 4.

Question 26: A chord is 8 units long and is 3 units from the center. What is the radius (to nearest whole)?

4
7
6
5 (Correct answer)

Correct answer: 5

Half-chord 4, distance 3: r = √(4² + 3²) = √25 = 5.

Question 27: The point on the unit circle at angle 90° has coordinates:

(−1, 0)
(0, −1)
(1, 0)
(0, 1) (Correct answer)

Correct answer: (0, 1)

At 90° the unit-circle point is (cos 90°, sin 90°) = (0, 1).

Question 28: A sector has a central angle of 90° in a circle of radius 4. What is its area?

4π (Correct answer)
8π
π
16π

Correct answer: 4π

Sector area = (90/360) × π(4²) = (1/4)(16π) = 4π.

Question 29: Two similar triangles have areas in the ratio 25:49. Their perimeters are in the ratio:

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

Correct answer: 5:7

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

Question 30: What does it mean for two shapes to be similar?

Same size, different shape.
Same shape, same size.
Same shape, sides proportional (Correct answer)
Completely identical.

Correct answer: Same shape, sides proportional

Two shapes are similar if they have the same shape but not necessarily the same size. This means their corresponding angles are equal, and the lengths of their corresponding sides are in proportion. The constant ratio between corresponding sides is known as the scale factor.

Question 31: How many degrees are in each exterior angle of a regular octagon?

30°
135°
60°
45° (Correct answer)

Correct answer: 45°

Exterior angles sum to 360°, so 360° ÷ 8 = 45°.

Question 32: What does it mean to 'construct' a geometric figure?

To create it using only a compass and straightedge following defined rules (Correct answer)
To model it in a 3D program
To measure and draw it using a ruler and protractor
To draw it freehand as accurately as possible

Correct answer: To create it using only a compass and straightedge following defined rules

A geometric construction uses only a compass and straightedge (no measurements), following classical rules.

Question 33: How to calculate a triangle's area

A = 2 × b × h
A = b × h
A = 1/2 × b × h (Correct answer)
A = 1/4 × b × h

Correct answer: A = 1/2 × b × h

The standard formula for calculating the area of any triangle is A = 1/2 × base × height. The base is any side of the triangle, and the height is the perpendicular distance from that base to the opposite vertex. This formula works for all types of triangles.

Question 34: A triangle whose single angle is more than 90 degrees.

Obtuse triangle (Correct answer)
Isosceles triangle
Acute triangle
Equilateral triangle

Correct answer: Obtuse triangle

An obtuse triangle is defined as a triangle that has one angle measuring more than 90 degrees. Since the sum of angles in any triangle is 180 degrees, a triangle can have at most one obtuse angle. The other two angles in an obtuse triangle must be acute (less than 90 degrees).

Question 35: What is the distance between points (1, 2) and (4, 6)?

7
5 (Correct answer)
3
25

Correct answer: 5

Distance = sqrt((4-1)^2 + (6-2)^2) = sqrt(9+16) = sqrt(25) = 5.

Question 36: Two similar cones have radii 3 and 5. What is the ratio of their surface areas?

3:5
27:125
6:10
9:25 (Correct answer)

Correct answer: 9:25

Surface area ratio is the square of the linear ratio: 3²:5² = 9:25.

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%

To Pass

Checking for saved progress...

Exam Rules

You can skip questions and return to them later
Flag questions for review before submitting
No feedback shown until you submit the entire exam
Unanswered questions count as wrong — answer everything
10 pretest questions are mixed in and don't affect your score
Timer auto-submits when time runs out
Your progress is auto-saved every 30 seconds