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

Consider the following: AB is a diameter of a circle with centre O. Radius OP is perpendicular to AB. Let Q be any point on arc PB. What is ∠BAP equal to?

  1. 30°

  2. 40°

  3. 45°

  4. 60°
Reveal answer Fill a bubble to check yourself
C Correct answer
Explanation

AB is diameter, O is center. OP is perpendicular to AB, so angle AOP = 90 degrees. Triangle OAP is an isosceles right triangle (OA=OP=radius). Thus, angle OAP = 45 degrees. Since O, A, B are collinear, angle BAP is the same as angle OAP, which is 45 degrees.

Multiple choice

Directions : Select the correct alternative from the given choices. Find the distance (in cm) between two parallel chords, drawn one on each side of the center of a circle of radius 65 cm, if their lengths are 104 cm and 120 cm respectively.

  1. 14

  2. 30

  3. 60

  4. 64
Reveal answer Fill a bubble to check yourself
D Correct answer
Explanation

Distance from center to chord of length 104 is sqrt(65^2 - 52^2) = sqrt(4225 - 2704) = sqrt(1521) = 39. Distance from center to chord of length 120 is sqrt(65^2 - 60^2) = sqrt(4225 - 3600) = sqrt(625) = 25. Since they are on opposite sides, the distance between them is 39 + 25 = 64.

Multiple choice

Two circles of radii 17 cm and 8 cm are concentric. The length of a chord of greater circle which touches the smaller circle is

  1. 15 cm

  2. 16 cm

  3. 30 cm

  4. 34 cm
Reveal answer Fill a bubble to check yourself
C Correct answer
Explanation

The chord of the larger circle tangent to the smaller circle forms a right triangle with the radius of the smaller circle (8) and the radius of the larger circle (17). Half the chord length is sqrt(17^2 - 8^2) = sqrt(289 - 64) = sqrt(225) = 15. The full chord length is 2 * 15 = 30 cm.

Multiple choice

The vertices A, B and C of a triangle are (2, 1), (5, 2) and (3, 4), respectively. What are the coordinates of circumcentre?

  1. (3.25, 2.25)

  2. (2, 6)

  3. (6.5, 4.5)

  4. (13, 9)

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

Circumcentre (x, y) is equidistant from (2, 1), (5, 2), (3, 4). (x-2)^2 + (y-1)^2 = (x-5)^2 + (y-2)^2 => -4x + 4 - 2y + 1 = -10x + 25 - 4y + 4 => 6x + 2y = 24 => 3x + y = 12. (x-2)^2 + (y-1)^2 = (x-3)^2 + (y-4)^2 => -4x + 4 - 2y + 1 = -6x + 9 - 8y + 16 => 2x + 6y = 20 => x + 3y = 10. Solving 3x+y=12 and x+3y=10: x = 3.25, y = 2.25.

Multiple choice

A circle C1 of radius b touches the circle x2 + y2 = a2 externally and has its centre on the positive x-axis. Another circle C2 of radius c touches the circle C1 externally and has its centre on the positive x-axis. If a < b < c, then the three circles have a common tangent if a, b, c are in

  1. A.P.

  2. G.P.

  3. H.P.

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

For three circles with radii a, b, c to have a common tangent, their radii must be in geometric progression (b^2 = ac).

Multiple choice

The locus of the point of intersection of the tangents at the extremities of a chord of the circle x2 + y2 = a2, which touches the circle x2 + y2 – 2ax = 0, is

  1. y2 = a(a – 2x)

  2. x2 = a(a – 2y)

  3. x2 + y2 = (x + a)2

  4. x2 + y2 = (y + a)2
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

The chord of contact of tangents from (h, k) to x^2 + y^2 = a^2 is hx + ky = a^2. This chord touches x^2 + y^2 - 2ax = 0 (center (a, 0), radius a). The distance from (a, 0) to the line hx + ky - a^2 = 0 is |ha - a^2| / sqrt(h^2 + k^2) = a. Squaring gives (a(h-a))^2 = a^2(h^2 + k^2), which simplifies to (h-a)^2 = h^2 + k^2, or h^2 - 2ah + a^2 = h^2 + k^2. Thus k^2 = a^2 - 2ah. Replacing (h, k) with (x, y) gives y^2 = a(a - 2x).

Multiple choice

The chord AB of a circle of centre O subtends an angle Θ with tangent at A to the circle. ∠ABO is?

  1. Θ

  2. 90° - Θ

  3. 90° + Θ

  4. 2(π - Θ)

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

By the alternate segment theorem, the angle between the tangent and the chord equals the angle in the alternate segment. If the angle between tangent and chord AB is theta, then the angle subtended by chord AB at the circumference is theta. The angle at the center is 2*theta. In triangle OAB, OA=OB (radii), so angle OAB = angle OBA = (180 - 2*theta) / 2 = 90 - theta.

Multiple choice

The radius of a circle is 5 cm. The distance of a point lying outside the circle from the centre is 13 cm. The length of the tangent drawn from the outside point to the circle is?

  1. 7 cm

  2. 9 cm

  3. 10 cm

  4. 12 cm

  5. 15 cm
Reveal answer Fill a bubble to check yourself
D Correct answer
Explanation

The radius, the tangent, and the line from the center to the external point form a right triangle where the distance to the center is the hypotenuse. 5^2 + tangent^2 = 13^2. 25 + tangent^2 = 169. tangent^2 = 144. tangent = 12.

Multiple choice

A circle C1 passes through the origin O and has diameter 4 on the positive x-axis. The line y = 2x gives a chord OA of a circle C1. Let C2 be the circle with OA as a diameter. If the tangent to C2 at the point A meets the x-axis at P and y-axis at Q, then QA : AP is equal to:

  1. 1 : 4

  2. 1 : 5

  3. 2 : 5

  4. 1 : 3
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

C1 has diameter 4 on x-axis, center (2,0), radius 2. Equation: (x-2)^2 + y^2 = 4. Line y=2x intersects C1 at A. Substituting: (x-2)^2 + 4x^2 = 4 -> x^2 - 4x + 4 + 4x^2 = 4 -> 5x^2 - 4x = 0. A is (4/5, 8/5). C2 has OA as diameter, center (2/5, 4/5), radius = sqrt((2/5)^2 + (4/5)^2) = sqrt(20/25) = sqrt(4/5). Tangent at A to C2 leads to the ratio 1:4.