Quantitative Aptitude

Geometry

1,950 Questions
All Circle Tangents and Chords 772 Coordinate Geometry 360 Circle and Arc Measures 216 Triangle Centers and Medians 175 Composite and Inscribed Shapes 174 Similarity, Congruence, and Proportionality 118 Right Triangles and Pythagorean Theorem 100 Standard 2D Mensuration 22 Polygon and Angle Sums 10 3D Solid Mensuration 3

Geometry Questions

Multiple choice

The ratio of the heights of two similar triangles is 3 : 4. If the area of the smaller triangle is 27 square centimeters, what is the area of the larger triangle?

  1. 36 square centimeters

  2. 45 square centimeters

  3. 48 square centimeters

  4. 64 square centimeters

  5. 72 square centimeters
Reveal answer Fill a bubble to check yourself
C Correct answer
Explanation

The ratio of the areas of two similar triangles is the square of the ratio of their corresponding sides. Since the ratio of sides is 3:4, the ratio of areas is 3^2:4^2, which is 9:16. Given the smaller area is 27, the larger area is (27/9) * 16 = 3 * 16 = 48.

Multiple choice

Two similar right triangles have corresponding side lengths in a ratio of 3 : 5. If the area of the smaller triangle is 54 square centimeters, what is the area of the larger triangle in square centimeters?

  1. 90

  2. 120

  3. 150

  4. 200
Reveal answer Fill a bubble to check yourself
C Correct answer
Explanation

The ratio of areas of similar triangles is the square of the ratio of their corresponding sides. Area ratio = (3/5)^2 = 9/25. Let A be the area of the larger triangle: 54 / A = 9 / 25. A = (54 * 25) / 9 = 6 * 25 = 150.

Multiple choice

A line segment AB of length 12 centimeters is tangent to a circle at point B. The distance from the center of the circle O to point A is 13 centimeters. What is the radius of the circle?

  1. 5 centimeters

  2. 6 centimeters

  3. 7 centimeters

  4. 8 centimeters
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

A tangent is perpendicular to the radius at the point of contact. This forms a right triangle with legs r and 12, and hypotenuse 13. By Pythagorean theorem, r^2 + 12^2 = 13^2, so r^2 + 144 = 169, r^2 = 25, r = 5.

Multiple choice

In a right-angled triangle ABC, angle A is 90 degrees. The lengths of sides AC and AB are 6 cm and 8 cm, respectively. A circle is inscribed within the triangle, with sides AB, BC, and AC as tangents to the inscribed circle. Another circle is circumscribed around the triangle, passing through points A, B, and C. Find the ratio of the area of the inscribed circle to the area of the circumscribed circle.

  1. 1 : 3

  2. 2 : 5

  3. 9 : 16

  4. 4 : 25

  5. a
Reveal answer Fill a bubble to check yourself
D Correct answer
Explanation

In a 6-8-10 triangle, the inradius r = (a+b-c)/2 = (6+8-10)/2 = 2. The circumradius R = hypotenuse/2 = 10/2 = 5. The ratio of areas is (pi*r^2) / (pi*R^2) = r^2 / R^2 = 2^2 / 5^2 = 4/25.

Multiple choice

Two circles, each with a radius of 7 cm, have a direct common tangent AB and an indirect common tangent CD. If the length of CD is 48 cm, find the length of AB.

  1. 25 cm

  2. 50 cm

  3. 55 cm

  4. 60 cm

  5. None of these
Reveal answer Fill a bubble to check yourself
B Correct answer
Explanation

For two circles of equal radius r, the length of the direct common tangent AB is equal to the distance between centers d. The length of the indirect common tangent CD is sqrt(d^2 - (2r)^2). Given CD = 48 and r = 7, 48^2 = d^2 - 14^2. 2304 = d^2 - 196. d^2 = 2500. d = 50.

Multiple choice

What is the area of the remaining portion of a circle of radius 14 cm when a square with the maximum possible area inscribed in it is removed from it? (Use ? = 22/7)

  1. 518 cm2

  2. 520 cm2

  3. 616 cm2

  4. 224 cm2
Reveal answer Fill a bubble to check yourself
D Correct answer
Explanation

Radius = 14. Area of circle = (22/7)*14*14 = 616. The largest inscribed square has a diagonal equal to the diameter (28). Side of square s: s*sqrt(2) = 28, so s = 28/sqrt(2) = 14*sqrt(2). Area of square = s^2 = 196*2 = 392. Remaining area = 616 - 392 = 224.