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Praxis Geometry study materials
Praxis Geometry study resources

Praxis Geometry practice questions

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Question 1Geometry content knowledge

In a circle with center O, chord AB has length 16 and its midpoint M is 6 units from O. What is the radius of the circle?

  1. 10 Correct answer.
  2. 2√7
  3. 12
  4. 8

Explanation

A radius drawn to the midpoint of a chord is perpendicular to that chord, so triangle OMA is right with legs 8 and 6. The radius is the hypotenuse, √(64+36) = √100 = 10.

Question 2Geometry content knowledge

A triangle has interior angles measuring (2x + 10)°, (3x − 20)°, and (x + 40)°. What is the measure of the smallest angle?

  1. 75°
  2. 55° Correct answer.
  3. 65°
  4. 60°

Explanation

The angles of a triangle sum to 180°, so 6x + 30 = 180, giving x = 25. The three angles are 60°, 55°, and 65°, so the smallest is 55°; checking each expression confirms 3(25) − 20 = 55.

Question 3Geometry content knowledge

Quadrilateral PQRS has diagonals that bisect each other and are congruent but not perpendicular. Which type of quadrilateral must PQRS be?

  1. Trapezoid
  2. Kite
  3. Rhombus
  4. Rectangle Correct answer.

Explanation

Diagonals that bisect each other make the figure a parallelogram, and congruent diagonals make that parallelogram a rectangle. Perpendicular diagonals would be needed to force a rhombus, which is why that classification is excluded.

Question 4Geometry content knowledge

A line passes through (2, −3) and is perpendicular to the line 4x − 6y = 12. What is the slope of this line?

  1. 3/2
  2. 2/3
  3. −2/3
  4. −3/2 Correct answer.

Explanation

Solving 4x − 6y = 12 for y gives y = (2/3)x − 2, so the given line has slope 2/3. A perpendicular line has the negative reciprocal slope, which is −3/2.

Question 5Geometry content knowledge

A regular hexagon is inscribed in a circle of radius 6. What is the area of the hexagon?

  1. 54√3 Correct answer.
  2. 36√3
  3. 27√3
  4. 108√3

Explanation

A regular hexagon splits into six equilateral triangles whose side equals the radius, 6. Each triangle has area (√3/4)(36) = 9√3, and multiplying by six gives 54√3.

Question 6Geometry content knowledge

A right circular cone has radius 5 and slant height 13. What is its total surface area?

  1. 65π
  2. 120π
  3. 40π
  4. 90π Correct answer.

Explanation

Total surface area of a cone is πr² + πrℓ. Substituting r = 5 and ℓ = 13 gives 25π + 65π = 90π.

Question 7Geometry content knowledge

Two similar triangles have areas in the ratio 25 : 49. What is the ratio of their corresponding side lengths?

  1. 5 : 49
  2. 5 : 7 Correct answer.
  3. 7 : 5
  4. 25 : 49

Explanation

For similar figures, the ratio of areas equals the square of the ratio of corresponding lengths. Taking the square root of 25/49 gives the length ratio 5/7.

Question 8Geometry content knowledge

Point A is at (−4, 1). After a rotation of 90° counterclockwise about the origin, what are the coordinates of the image of A?

  1. (−1, 4)
  2. (1, 4)
  3. (−1, −4) Correct answer.
  4. (4, −1)

Explanation

A 90° counterclockwise rotation about the origin maps (x, y) to (−y, x). Applying this to (−4, 1) gives (−1, −4).

Question 9Geometry content knowledge

A sphere has a volume of 288π cubic units. What is its radius?

  1. 4
  2. 9
  3. 12
  4. 6 Correct answer.

Explanation

The volume formula is V = (4/3)πr³. Setting (4/3)πr³ = 288π gives r³ = 216, so r = 6.

Question 10Geometry content knowledge

In right triangle ABC, the right angle is at C, AB = 13, and BC = 5. What is the value of sin A?

  1. 12/13
  2. 5/13 Correct answer.
  3. 13/5
  4. 5/12

Explanation

Side AC = √(169 − 25) = 12. Angle A is opposite side BC, so sin A = opposite/hypotenuse = 5/13.

Question 11Geometry content knowledge

A line segment has endpoints (1, 2) and (7, 10). What are the coordinates of its midpoint?

  1. (8, 12)
  2. (4, 6) Correct answer.
  3. (6, 8)
  4. (3, 4)

Explanation

The midpoint formula averages the x-coordinates and the y-coordinates separately. Averaging gives ((1+7)/2, (2+10)/2) = (4, 6).

Question 12Geometry content knowledge

Two parallel lines are cut by a transversal. One interior angle on the left side of the transversal measures 3x + 15, and the same-side interior angle measures 2x + 25. What is the value of x?

  1. 28 Correct answer.
  2. 20
  3. 35
  4. 40

Explanation

Same-side interior angles formed by parallel lines are supplementary, so (3x + 15) + (2x + 25) = 180. This gives 5x + 40 = 180, so x = 28.

Question 13Geometry content knowledge

A dilation centered at the origin maps point (3, −6) to (12, −24). What is the scale factor?

  1. 4 Correct answer.
  2. 1/4
  3. 3
  4. −4

Explanation

A dilation about the origin multiplies both coordinates by the same scale factor k. Since 12/3 = 4 and −24/−6 = 4, the scale factor is 4.

Question 14Geometry content knowledge

A parallelogram has vertices at (0, 0), (6, 0), (8, 5), and (2, 5). What is its area?

  1. 24
  2. 40
  3. 30 Correct answer.
  4. 48

Explanation

The base along the x-axis has length 6, and the vertical distance between the horizontal sides is 5. Area equals base times height, 6 × 5 = 30.

Question 15Geometry content knowledge

A regular polygon has interior angles measuring 156° each. How many sides does the polygon have?

  1. 18
  2. 10
  3. 12
  4. 15 Correct answer.

Explanation

Each exterior angle is 180° − 156° = 24°, and exterior angles of a convex polygon sum to 360°. Dividing 360 by 24 gives 15 sides.

Question 16Geometry content knowledge

Triangle ABC is congruent to triangle DEF under a correspondence that maps A to D, B to E, and C to F. If AB = 3x − 4 and DE = x + 8, what is the length of AB?

  1. 6
  2. 10
  3. 18
  4. 14 Correct answer.

Explanation

Corresponding parts of congruent triangles are equal, so 3x − 4 = x + 8, giving 2x = 12 and x = 6. Substituting gives AB = 3(6) − 4 = 14.

Question 17Geometry content knowledge

A rectangular prism has dimensions 3, 4, and 12. What is the length of its space diagonal?

  1. 19
  2. √151
  3. 15
  4. 13 Correct answer.

Explanation

The space diagonal of a box is √(a² + b² + c²). Here √(9 + 16 + 144) = √169 = 13.

Question 18Geometry content knowledge

A triangle has side lengths 7, 24, and 25. Which classification is correct?

  1. Acute triangle
  2. Isosceles triangle
  3. Right triangle Correct answer.
  4. Obtuse triangle

Explanation

Check the Pythagorean relationship with the longest side: 7² + 24² = 49 + 576 = 625 = 25². Since the squares satisfy the theorem, the triangle is right.

Question 19Geometry content knowledge

In the coordinate plane, a circle has equation x² + y² − 6x + 8y = 0. What is the radius of the circle?

  1. 5 Correct answer.
  2. 10
  3. 25
  4. √7

Explanation

Complete the square: (x² − 6x + 9) + (y² + 8y + 16) = 25, so (x − 3)² + (y + 4)² = 25. The radius is √25 = 5.

Question 20Geometry content knowledge

A 10-foot ladder leans against a vertical wall, with its base 6 feet from the wall. If the base slides 2 feet farther from the wall, how far does the top of the ladder move down the wall?

  1. 4 feet
  2. 1 foot
  3. 3 feet
  4. 2 feet Correct answer.

Explanation

Initially the top is at √(100 − 36) = 8 feet. After the base moves to 8 feet, the top is at √(100 − 64) = 6 feet. The top descends 8 − 6 = 2 feet.

Question 21Geometry content knowledge

A triangle has vertices at A(1, 1), B(5, 1), and C(1, 4). The triangle is dilated by a factor of 3 centered at the origin, then translated 2 units left. What is the area of the final image?

  1. 18 Correct answer.
  2. 12
  3. 6
  4. 54

Explanation

The original right triangle has legs 4 and 3, so its area is 6. Dilation by 3 multiplies area by 9, giving 54, while translation does not change area; the final area is 54.

Question 22Geometry content knowledge

Two chords of a circle intersect inside the circle. One chord is divided into segments of length 4 and 9. The other chord is divided into segments where one part is 6. What is the length of the other part?

  1. 3
  2. 4
  3. 9
  4. 6 Correct answer.

Explanation

When two chords intersect inside a circle, the products of their segment lengths are equal. So 4 × 9 = 36, and 36 ÷ 6 = 6.

Question 23Geometry content knowledge

A cone and a cylinder share the same radius and height. If the cylinder's volume is 81π cubic units, what is the cone's volume?

  1. 40.5π
  2. 27π Correct answer.
  3. 243π
  4. 81π

Explanation

The volume of a cone is one-third of the volume of a cylinder with the same base and height. One-third of 81π is 27π.

Question 24Geometry content knowledge

A quadrilateral has opposite angles that are supplementary. Which conclusion is valid?

  1. The quadrilateral is a rhombus.
  2. The quadrilateral is a rectangle.
  3. The quadrilateral is a parallelogram.
  4. The quadrilateral is cyclic, so it can be inscribed in a circle. Correct answer.

Explanation

A quadrilateral is cyclic exactly when a pair of opposite angles is supplementary, which is the converse of the cyclic quadrilateral angle theorem. This condition does not force any particular side relationships.

Question 25Geometry content knowledge

In a circle, an inscribed angle intercepts an arc measuring 84°. What is the measure of the inscribed angle?

  1. 84°
  2. 168°
  3. 21°
  4. 42° Correct answer.

Explanation

The measure of an inscribed angle equals half the measure of its intercepted arc. Here the arc is 84°, so the angle is 84° ÷ 2 = 42°.

Question 26Geometry content knowledge

A tangent segment from an external point touches a circle of radius 9. The external point is 12 units from the circle's center. What is the length of the tangent segment?

  1. 21
  2. 3
  3. 15
  4. √63 Correct answer.

Explanation

A tangent is perpendicular to the radius at the point of tangency, forming a right triangle with legs 9 and t and hypotenuse 12. By the Pythagorean theorem, t² = 144 − 81 = 63, so t = √63.

Question 27Geometry content knowledge

A sector of a circle of radius 10 has a central angle of 72°. What is the arc length of the sector?

  1. 2π
  2. 8π
  3. 20π
  4. 4π Correct answer.

Explanation

Arc length equals the fraction of the circle times the circumference: (72/360)(2π·10) = (1/5)(20π) = 4π.

Question 28Geometry content knowledge

A cylinder has radius 4 and height 10. What is its lateral surface area?

  1. 160π
  2. 104π
  3. 40π
  4. 80π Correct answer.

Explanation

The lateral surface area of a cylinder equals 2πrh, the circumference times the height. Substituting gives 2π(4)(10) = 80π.

Question 29Geometry content knowledge

A triangular prism has a base that is a right triangle with legs 6 and 8, and the prism is 5 units long. What is its volume?

  1. 60
  2. 120 Correct answer.
  3. 480
  4. 240

Explanation

The base area of the right triangle is (1/2)(6)(8) = 24. Multiplying by the prism length gives volume 24·5 = 120 cubic units.

Question 30Geometry content knowledge

A pyramid has a square base with side length 6 and height 4. What is its volume?

  1. 24
  2. 72
  3. 144
  4. 48 Correct answer.

Explanation

Volume of a pyramid is one-third of the base area times the height. The base area is 36, so V = (1/3)(36)(4) = 48 cubic units.

Question 31Geometry content knowledge

Two similar solids have corresponding edge lengths in the ratio 3 : 5. What is the ratio of their surface areas?

  1. 3 : 5
  2. 27 : 125
  3. 9 : 25 Correct answer.
  4. 6 : 10

Explanation

For similar solids, surface areas scale as the square of the linear ratio. Squaring 3 : 5 gives 9 : 25.

Question 32Geometry content knowledge

A circle has a radius of 8. A chord is 3 units from the center. What is the length of the chord?

  1. √73
  2. √55
  3. 2√73
  4. 2√55 Correct answer.

Explanation

The perpendicular from the center to a chord bisects it. The half-chord, the distance 3, and radius 8 form a right triangle, so half-chord = √(64 − 9) = √55, making the full chord 2√55.

Question 33Geometry content knowledge

In a circle, a secant and a tangent meet at an external point. The tangent segment is 6 and the external part of the secant is 4. What is the length of the entire secant?

  1. 8
  2. 9 Correct answer.
  3. 24
  4. 12

Explanation

The tangent-secant theorem gives tangent² = external part × whole secant. Thus 36 = 4 × whole, so the whole secant is 9.

Question 34Geometry content knowledge

A quadrilateral inscribed in a circle has one angle measuring 115°. What is the measure of the opposite angle?

  1. 65° Correct answer.
  2. 25°
  3. 230°
  4. 115°

Explanation

Opposite angles of a cyclic quadrilateral are supplementary, summing to 180°. Therefore the opposite angle is 180° − 115° = 65°.

Question 35Geometry content knowledge

A line passes through (2, 5) and is perpendicular to a line with slope 4. What is the slope of this line?

  1. −1/4 Correct answer.
  2. 4
  3. 1/4
  4. −4

Explanation

Perpendicular lines have slopes that are negative reciprocals. The negative reciprocal of 4 is −1/4.

Question 36Geometry content knowledge

A line segment has endpoints (−2, 3) and (4, 7). What is the slope of the perpendicular bisector of this segment?

  1. 2/3
  2. −2/3
  3. 3/2
  4. −3/2 Correct answer.

Explanation

The segment's slope is (7 − 3)/(4 − (−2)) = 4/6 = 2/3. The perpendicular bisector has the negative reciprocal slope, −3/2.

Question 37Geometry content knowledge

A circle has center (2, −1) and passes through the point (5, 3). What is the standard equation of the circle?

  1. (x − 2)² + (y + 1)² = 25 Correct answer.
  2. (x − 2)² + (y + 1)² = 5
  3. (x − 2)² + (y − 1)² = 25
  4. (x + 2)² + (y − 1)² = 25

Explanation

The radius squared is (5 − 2)² + (3 + 1)² = 9 + 16 = 25. A circle with center (h, k) has equation (x − h)² + (y − k)² = r², giving (x − 2)² + (y + 1)² = 25.

Question 38Geometry content knowledge

The equation x² + y² + 4x − 10y + 20 = 0 represents a circle. What is its center?

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

Explanation

Completing the square: (x² + 4x + 4) + (y² − 10y + 25) = −20 + 4 + 25, giving (x + 2)² + (y − 5)² = 9. The center is (−2, 5).

Question 39Geometry content knowledge

Triangle PQR has vertices P(0, 0), Q(4, 0), and R(0, 3). It is reflected over the line y = x. What are the coordinates of the image of R?

  1. (0, −3)
  2. (3, 0) Correct answer.
  3. (−3, 0)
  4. (0, 3)

Explanation

Reflection over y = x swaps the x- and y-coordinates of each point. Applying this to R(0, 3) gives the image (3, 0).

Question 40Geometry content knowledge

A figure is rotated 180° about the origin. Which transformation produces the same image as this rotation?

  1. A reflection over the origin Correct answer.
  2. A reflection over the line y = x
  3. A reflection over the y-axis
  4. A reflection over the x-axis

Explanation

A 180° rotation about the origin maps (x, y) to (−x, −y), which is exactly the result of a reflection through the origin (a point reflection).

Question 41Geometry content knowledge

A regular octagon is inscribed in a circle. What is the measure of each central angle formed by consecutive vertices?

  1. 22.5°
  2. 45° Correct answer.
  3. 135°
  4. 60°

Explanation

A regular polygon with n sides divides the full 360° of the circumscribed circle into n equal central angles. For an octagon, 360° ÷ 8 = 45°.

Question 42Geometry content knowledge

A regular pentagon has side length 8 and apothem approximately 5.5. What is its area?

  1. 44
  2. 220
  3. 88
  4. 110 Correct answer.

Explanation

The area of a regular polygon is (1/2)(apothem)(perimeter). The perimeter is 5 × 8 = 40, so the area is (1/2)(5.5)(40) = 110.

Question 43Geometry content knowledge

In a 30°-60°-90° triangle, the hypotenuse has length 14. What is the length of the side opposite the 60° angle?

  1. 7√3 Correct answer.
  2. 7
  3. 14√3
  4. 7√2

Explanation

In a 30°-60°-90° triangle, the sides are in ratio 1 : √3 : 2. The short leg is half the hypotenuse, 7, and the long leg opposite 60° is 7√3.

Question 44Geometry content knowledge

A right triangle has legs of length 5 and 12. What is the length of the altitude drawn to the hypotenuse?

  1. 13/60
  2. 60/13 Correct answer.
  3. 30/13
  4. 12/5

Explanation

The hypotenuse is √(25 + 144) = 13. The altitude to the hypotenuse equals (leg₁ × leg₂)/hypotenuse = (5 × 12)/13 = 60/13.

Question 45Geometry content knowledge

The geometric mean of two segments of the hypotenuse created by an altitude is used to find the altitude. If the two segments measure 4 and 9, what is the altitude?

  1. 13
  2. 36
  3. 6 Correct answer.
  4. 6.5

Explanation

The altitude to the hypotenuse is the geometric mean of the two segments it creates: h = √(4 × 9) = √36 = 6.

Question 46Geometry content knowledge

A triangle has side lengths 9, 12, and 15. Which statement about the triangle is true?

  1. It is obtuse.
  2. It is right. Correct answer.
  3. It is impossible to form.
  4. It is acute.

Explanation

Check the Pythagorean relationship: 9² + 12² = 81 + 144 = 225 = 15². Since the squares of the two shorter sides sum to the square of the longest, the triangle is right.

Question 47Geometry content knowledge

In a coordinate proof, a quadrilateral has diagonals that bisect each other and are congruent. Which type of quadrilateral must it be?

  1. A rhombus
  2. A trapezoid
  3. A rectangle Correct answer.
  4. A kite

Explanation

If the diagonals of a quadrilateral bisect each other, it is a parallelogram. If, in addition, the diagonals are congruent, the parallelogram is a rectangle.

Question 48Geometry content knowledge

A quadrilateral has diagonals that are perpendicular and bisect each other. Which conclusion about the quadrilateral is valid?

  1. It is a rhombus. Correct answer.
  2. It is a rectangle.
  3. It is a square.
  4. It is an isosceles trapezoid.

Explanation

Diagonals that bisect each other make the figure a parallelogram, and perpendicular diagonals in a parallelogram force all four sides to be congruent, giving a rhombus. Congruent diagonals would be needed to guarantee a square.

Question 49Geometry content knowledge

A triangle has two sides of length 7 and 11. Which value could be the length of the third side?

  1. 3
  2. 17 Correct answer.
  3. 18
  4. 4

Explanation

The triangle inequality requires the third side to be greater than 11 − 7 = 4 and less than 11 + 7 = 18. Of the choices, only 17 lies strictly between 4 and 18.

Question 50Geometry content knowledge

In a plane, two distinct lines are cut by a transversal. Which condition guarantees that the two lines are parallel?

  1. A pair of consecutive interior angles are congruent
  2. A pair of alternate exterior angles are supplementary
  3. A pair of corresponding angles are supplementary
  4. A pair of alternate interior angles are congruent Correct answer.

Explanation

Alternate interior angles lie on opposite sides of the transversal between the two lines. When these angles are congruent, the lines must be parallel, which is the converse of the alternate interior angles theorem. The other angle pairings listed give relationships that do not force the lines to be parallel.