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

A square is inscribed in a circle. If the side of the square is 14 cm, what is the area (in sq. cm) of the circle?

  1. 49π

  2. 77π

  3. 121π

  4. 98π
Reveal answer Fill a bubble to check yourself
D Correct answer
Explanation

A square inscribed in a circle has its diagonal equal to the diameter of the circle. For a square with side 14 cm, the diagonal is 14 * sqrt(2) cm. The radius is half the diameter, so r = 7 * sqrt(2) cm. The area of the circle is pi * r^2 = pi * (7 * sqrt(2))^2 = pi * 49 * 2 = 98 * pi.

Multiple choice

The sum of the squares of the lengths of the chords intercepted on the circle, x2 + y2 = 16, by the lines, x + y = n, n ∈ N, where N is the set of all natural numbers, is

  1. 320

  2. 160

  3. 105

  4. 210
Reveal answer Fill a bubble to check yourself
D Correct answer
Explanation

The chord length L of x^2 + y^2 = r^2 cut by x + y = n is 2 * sqrt(r^2 - d^2), where d is the distance from origin to line. Here r=4, d=n/sqrt(2). L^2 = 4(16 - n^2/2) = 64 - 2n^2. Summing for n=1 to 5 (since n^2 < 32): (64-2) + (64-8) + (64-18) + (64-32) + (64-50) = 62 + 56 + 46 + 32 + 14 = 210.

Multiple choice

The radii of two circles are 4.5 cm and 3.5 cm, respectively. The distance between the centres of the circles is 10 cm. What is the length of the transverse common tangent?

  1. 4 cm

  2. 5 cm

  3. 6 cm

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

The length of the transverse common tangent is given by the formula sqrt(d^2 - (r1 + r2)^2), where d is the distance between centers and r1, r2 are the radii. Here, d = 10, r1 = 4.5, and r2 = 3.5, so r1 + r2 = 8. The calculation is sqrt(10^2 - 8^2) = sqrt(100 - 64) = sqrt(36) = 6 cm.

Multiple choice

A chord of length 60 cm is at a distance of 16 cm from the centre of a circle. What is the radius (in cm) of the circle?

  1. 17

  2. 34

  3. 51

  4. 68
Reveal answer Fill a bubble to check yourself
B Correct answer
Explanation

A perpendicular from the center to a chord bisects the chord. This creates a right triangle with the radius as the hypotenuse, half the chord (30 cm) as one leg, and the distance from the center (16 cm) as the other leg. Radius = sqrt(30^2 + 16^2) = sqrt(900 + 256) = sqrt(1156) = 34.

Multiple choice

If the equation of the locus of the mid-point of the chord which subtends right angle at the centre of a circle is x2 + y2 = 9, then find the equation of the circle on which this chord exists.

  1. x2 + y2 = 36

  2. 2x2 + 2y2 = 9

  3. x2 + y2 = 18

  4. (x - 1)2 + (y - 2)2 = 9

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

For a chord of a circle to subtend a right angle at the center, the distance from the center to the chord must be r/sqrt(2). If the locus of the midpoint is x^2 + y^2 = 9, the radius of this locus circle is 3. Since the distance from the center to the chord is 3, the radius of the original circle is 3 * sqrt(2). Thus, the equation is x^2 + y^2 = (3 * sqrt(2))^2 = 18.

Multiple choice

If the tangent at (1, 7) to the curve x2 = y - 6 touches the circle x2 + y2 + 16x + 12y + c = 0, then the value of c is

  1. 195

  2. 185

  3. 85

  4. 95
Reveal answer Fill a bubble to check yourself
D Correct answer
Explanation

Curve x^2 = y - 6 => y = x^2 + 6. Derivative dy/dx = 2x. At (1, 7), slope m = 2. Tangent: y - 7 = 2(x - 1) => y = 2x + 5 => 2x - y + 5 = 0. Circle x^2 + y^2 + 16x + 12y + c = 0 has center (-8, -6) and radius sqrt(64 + 36 - c) = sqrt(100 - c). Distance from center to tangent = |2(-8) - (-6) + 5| / sqrt(2^2 + (-1)^2) = |-16 + 6 + 5| / sqrt(5) = 5 / sqrt(5) = sqrt(5). Radius^2 = 5 = 100 - c => c = 95.

Multiple choice

Radius of a circle is 6 cm. The distance of a point lying outside the circle from the centre is 10 cm. The length of tangent drawn from the outside point to the circle is

  1. 5 cm

  2. 6 cm

  3. 7 cm

  4. 8 cm
Reveal answer Fill a bubble to check yourself
D Correct answer
Explanation

The radius, the tangent, and the distance from the center form a right-angled triangle where the distance from the center is the hypotenuse. Using Pythagoras theorem: tangent^2 + 6^2 = 10^2, so tangent^2 = 100 - 36 = 64, giving a length of 8 cm.

Multiple choice

A circle is inscribed in a triangle ABC. It touches the sides AB, BC and AC at the points R, P and Q, respectively. If AQ = 4.5 cm, PC = 5.5 cm and BR = 6 cm, then the perimeter of the triangle ABC is

  1. 30.5 cm

  2. 28 cm

  3. 32 cm

  4. 26.5 cm
Reveal answer Fill a bubble to check yourself
C Correct answer
Explanation

Tangents from a point to a circle are equal. AQ = AR = 4.5. PC = PQ = 5.5. BR = BP = 6. Sides are AB = 4.5+6 = 10.5, BC = 6+5.5 = 11.5, AC = 4.5+5.5 = 10. Perimeter = 10.5 + 11.5 + 10 = 32.