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

If the radius of the circle is 15 cm and length of the one of the cords is 24 cm, find the distance of midpoint of the chord from the centre of the circle.

  1. 9 cm

  2. 12 cm

  3. 15 cm

  4. 18 cm

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

A perpendicular from the center to a chord bisects the chord. This forms a right triangle with the radius (15) as the hypotenuse and half the chord (12) as one leg. The distance from the center is the other leg: sqrt(15^2 - 12^2) = sqrt(225 - 144) = sqrt(81) = 9 cm.

Multiple choice

Consider a circle of diameter 60 m with a chord of length 30 m. Compute the length of the minor arc on one side of the chord and the length of the major arc.

  1. 31.42 and 151.1

  2. 31.42 and 157.1

  3. 31.11 and 151.7

  4. 31.11 and 151.1

  5. 30.21 and 149.36
Reveal answer Fill a bubble to check yourself
B Correct answer
Explanation

Radius r = 30. Chord length = 30. This forms an equilateral triangle with the center, so the central angle is 60 degrees (pi/3 radians). Minor arc = r * theta = 30 * (pi/3) = 10 * pi = 31.4159. Major arc = 2 * pi * r - 31.4159 = 188.495 - 31.4159 = 157.08.

Multiple choice

PA and PB are two tangents to a circle with centre O. If ∠AOP : ∠OPB = 13 : 5, find the measure of ∠OPA.

  1. 20°

  2. 25°

  3. 45°

  4. 15°
Reveal answer Fill a bubble to check yourself
B Correct answer
Explanation

In triangle OAP, angle OAP = 90 degrees. Given angle AOP : angle OPB = 13:5. Since PA and PB are tangents, angle AOP = angle BOP. Thus, angle AOP : angle OPA = 13:5 is not the ratio, but angle AOP = 13x and angle OPA = 5x. In triangle OAP, 13x + 90 + 5x = 180, so 18x = 90, x = 5. Angle OPA = 5 * 5 = 25 degrees.

Multiple choice

Directions: In the following question, we have two quantities given. Compare these quantities and answer accordingly with the following given options. Quantity I: The area of the circle inscribed in a square of side 28 cm. Quantity II: The area of the circle circumscribing a rectangle of length and breadth equal to 21 cm and 20 cm, respectively.

  1. Quantity I > Quantity II

  2. Quantity I < Quantity II

  3. Quantity I ≥ Quantity II

  4. Quantity II ≥ Quantity I

  5. Quantity I = Quantity II or No relation
Reveal answer Fill a bubble to check yourself
B Correct answer
Explanation

Quantity I: Circle inscribed in square of side 28. Radius = 14. Area = pi * 14^2 = 196 * pi. Quantity II: Circle circumscribing rectangle 21x20. Diameter = sqrt(21^2 + 20^2) = sqrt(441 + 400) = sqrt(841) = 29. Radius = 14.5. Area = pi * 14.5^2 = 210.25 * pi. Quantity II > Quantity I.

Multiple choice

If in a right-angled triangle, the hypotenuse is to be 1 cm longer than the base and 2 cm longer than the altitude, then the perimeter of the triangle will be

  1. 24 cm

  2. 20 cm

  3. 12 cm

  4. 10 cm
Reveal answer Fill a bubble to check yourself
C Correct answer
Explanation

Let hypotenuse be h, base b, altitude a. h = b+1, h = a+2. So b = h-1, a = h-2. By Pythagoras: (h-1)^2 + (h-2)^2 = h^2. h^2 - 2h + 1 + h^2 - 4h + 4 = h^2. h^2 - 6h + 5 = 0. (h-5)(h-1) = 0. h=5 (h=1 is impossible). Sides are 5, 4, 3. Perimeter = 5+4+3 = 12.