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Geometry

Mathematical Abilities — Paper-I · Official 2026 Coverage

Useful for: SSC CGL

Overview

Tier-II geometry tests triangle centres, similarity, angle bisectors and circle theorems in combination.

Key points

Right triangle inradius r = (a + b − c)/2; circumradius R = hypotenuse/2.
Angle bisector theorem: the bisector from A divides BC in the ratio AB : AC.
Intersecting chords: AP × PB = CP × PD.
Tangent–secant: (tangent)² = external segment × whole secant.
In a cyclic quadrilateral, opposite angles sum to 180°.

Explanation

Similar triangles: corresponding sides proportional; areas in the ratio of squares.
Mid-point theorem: the segment joining mid-points of two sides is parallel to and half the third side.
Centroid divides medians 2:1; area of the triangle formed by joining mid-points = 1/4 of the original.
Exterior angle of a cyclic quadrilateral equals the interior opposite angle.

Step by step

Identify the concept and the formula that links the given values.
Solve step by step, keeping units consistent.
Check the answer against the options and by a quick estimate.

Examples

Triangle with sides 6, 8, 10: right-angled; inradius = (6 + 8 − 10)/2 = 2; circumradius = 5.
AB = 6, AC = 9, BC = 10; the bisector from A meets BC at D → BD : DC = 6 : 9 → BD = 4.
Chords intersect at P with AP = 4, PB = 6, CP = 3 → PD = 24/3 = 8.

Traps and advanced points

Trap: Using side ratio for area ratio.
Trap: Applying the bisector theorem to a median.
Trap: Mixing tangent–secant and chord formulas.

Quick revision

r = (a + b − c)/2 for right triangles.
Bisector: AB : AC.
AP·PB = CP·PD.

Practice MCQs

Two angles of a triangle are 64° and 68°. Find the third angle.
1. 53°
2. 58°
3. 43°
4. 48°
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Answer: (d) 48°. Sum of angles in a triangle = 180°.
Using π=22/7, find the circumference of a circle of radius 385 cm.
1. 2464 cm
2. 2420 cm
3. 2398 cm
4. 2442 cm
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Answer: (b) 2420 cm. Circumference = 2πr.
Two angles of a triangle are 79° and 53°. Find the third angle.
1. 53°
2. 43°
3. 48°
4. 58°
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Answer: (c) 48°. Sum of angles in a triangle = 180°.
Using π=22/7, find the circumference of a circle of radius 63 cm.
1. 396 cm
2. 374 cm
3. 418 cm
4. 440 cm
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Answer: (a) 396 cm. Circumference = 2πr.
Two angles of a triangle are 26° and 37°. Find the third angle.
1. 117°
2. 112°
3. 122°
4. 127°
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Answer: (a) 117°. Sum of angles in a triangle = 180°.
Using π=22/7, find the circumference of a circle of radius 14 cm.
1. 132 cm
2. 88 cm
3. 66 cm
4. 110 cm
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Answer: (b) 88 cm. Circumference = 2πr.
Two angles of a triangle are 27° and 39°. Find the third angle.
1. 109°
2. 124°
3. 114°
4. 119°
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Answer: (c) 114°. Sum of angles in a triangle = 180°.
Using π=22/7, find the circumference of a circle of radius 21 cm.
1. 176 cm
2. 132 cm
3. 154 cm
4. 110 cm
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Answer: (b) 132 cm. Circumference = 2πr.
Two angles of a triangle are 28° and 41°. Find the third angle.
1. 111°
2. 121°
3. 106°
4. 116°
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Answer: (a) 111°. Sum of angles in a triangle = 180°.
Using π=22/7, find the circumference of a circle of radius 28 cm.
1. 220 cm
2. 176 cm
3. 198 cm
4. 154 cm
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Answer: (b) 176 cm. Circumference = 2πr.
Two angles of a triangle are 29° and 43°. Find the third angle.
1. 118°
2. 113°
3. 108°
4. 103°
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Answer: (c) 108°. Sum of angles in a triangle = 180°.
Using π=22/7, find the circumference of a circle of radius 35 cm.
1. 220 cm
2. 198 cm
3. 264 cm
4. 242 cm
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Answer: (a) 220 cm. Circumference = 2πr.
Two angles of a triangle are 30° and 45°. Find the third angle.
1. 105°
2. 115°
3. 110°
4. 100°
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Answer: (a) 105°. Sum of angles in a triangle = 180°.
Using π=22/7, find the circumference of a circle of radius 42 cm.
1. 286 cm
2. 308 cm
3. 264 cm
4. 242 cm
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Answer: (c) 264 cm. Circumference = 2πr.
Two angles of a triangle are 31° and 47°. Find the third angle.
1. 97°
2. 112°
3. 102°
4. 107°
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Answer: (c) 102°. Sum of angles in a triangle = 180°.