Geometry — Triangles, Circles & Quadrilaterals — Practice

Question 1

In △ABC, ∠A = 64°. I is the incentre. ∠BIC = ?
(a) 116°  (b) 122°  (c) 128°  (d) 132°

Accepted Answer

∠BIC = 90° + A/2 = 90° + 32° = 122°.

Answer: (b) 122°.

Question 2

G is the centroid of △ABC and AD is a median with GD = 5 cm. The length of AD is:
(a) 10 cm  (b) 12.5 cm  (c) 15 cm  (d) 20 cm

Accepted Answer

AG : GD = 2 : 1, so GD is one-third of AD.

AD = 3 × 5 = 15 cm.

Answer: (c) 15 cm.

Question 3

A point P is 25 cm from the centre O of a circle. The tangent from P has length 24 cm. The diameter of the circle is:
(a) 7 cm  (b) 12 cm  (c) 14 cm  (d) 15 cm

Accepted Answer

r = √(25² − 24²) = √(625 − 576) = √49 = 7 cm.

Diameter = 14 cm.

Answer: (c) 14 cm. (7–24–25 triple.)

Question 4

ABCD is a cyclic quadrilateral with ∠A = 3x°, ∠C = 2x°. ∠A equals:
(a) 72°  (b) 96°  (c) 108°  (d) 120°

Accepted Answer

Opposite angles of a cyclic quadrilateral sum to 180°: 3x + 2x = 180 → x = 36.

∠A = 108°.

Answer: (c) 108°.

Question 5

The areas of two similar triangles are 81 cm² and 144 cm². If the shorter triangle has a median of 10.8 cm, the corresponding median of the larger is:
(a) 12.6 cm  (b) 14.4 cm  (c) 16.2 cm  (d) 19.2 cm

Accepted Answer

Side (and median) ratio = √(81:144) = 9:12 = 3:4.

Larger median = 10.8 × 4/3 = 14.4 cm.

Answer: (b) 14.4 cm — every corresponding length scales with k, not just sides.

Question 6

A chord of length 30 cm is at distance 8 cm from the centre. The radius is:
(a) 15 cm  (b) 16 cm  (c) 17 cm  (d) 19 cm

Accepted Answer

(30/2)² + 8² = r² → 225 + 64 = 289 → r = 17 cm.

Answer: (c) 17 cm. (8–15–17 triple.)

Question 7

From an external point P, a tangent PT = 12 cm and a secant meets the circle at A and B with PA = 8 cm. Then AB equals:
(a) 8 cm  (b) 10 cm  (c) 12 cm  (d) 18 cm

Accepted Answer

PT² = PA × PB → 144 = 8 × PB → PB = 18.

AB = PB − PA = 18 − 8 = 10 cm.

Answer: (b) 10 cm. Trap: option (d) 18 is PB, not AB.

Question 8

Two circles of radii 8 cm and 3 cm have centres 13 cm apart. The length of the direct common tangent is:
(a) 10 cm  (b) 11 cm  (c) 12 cm  (d) √69 cm

Accepted Answer

Direct tangent = √(d² − (r₁ − r₂)²) = √(169 − 25) = √144 = 12 cm.

Answer: (c) 12 cm. (Transverse would use (r₁ + r₂)² = 121 → √48 — not asked.)

Question 9

In right △ABC (right angle at A), the altitude AD to hypotenuse BC gives BD = 4.5 cm and DC = 8 cm. AB equals:
(a) 6 cm  (b) 7.5 cm  (c) 9 cm  (d) 10 cm

Accepted Answer

AB² = BD × BC = 4.5 × 12.5 = 56.25 → AB = 7.5 cm.

Answer: (b) 7.5 cm. (Check: AD = √(4.5 × 8) = 6, AC = √(8 × 12.5) = 10 — the 7.5–10–12.5 triple.)

Question 10

Each interior angle of a regular polygon measures 156°. The number of sides is:
(a) 12  (b) 13  (c) 15  (d) 18

Accepted Answer

Exterior angle = 180 − 156 = 24°.

n = 360/24 = 15.

Answer: (c) 15.

Question 11

PQ is a tangent to a circle at A, and AB is a chord with ∠BAQ = 58°. C is a point on the major arc. ∠ACB equals:
(a) 58°  (b) 62°  (c) 116°  (d) 122°

Accepted Answer

By the alternate segment theorem, the tangent–chord angle equals the angle in the alternate segment: ∠ACB = ∠BAQ = 58°.

Answer: (a) 58°.

Question 12

The diagonals of a rhombus are 24 cm and 10 cm. Its perimeter is:
(a) 13 cm  (b) 40 cm  (c) 52 cm  (d) 60 cm

Accepted Answer

Half-diagonals 12 and 5 meet at 90° → side = √(144 + 25) = 13 cm.

Perimeter = 4 × 13 = 52 cm.

Answer: (c) 52 cm. (5–12–13 triple.)

Practice — Geometry — Triangles, Circles & Quadrilaterals

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