Geometry

Mathematics · Algebra, Geometry & Data

Useful for: SSC CGL, Railway RRB, CTET, OTET, OSSTET, UGC NET, OSSC RHT — Regular High School Teacher, UPSC CDS

and 21 more exams

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Overview

Most geometry items need a handful of angle, triangle and circle theorems applied in one or two steps.

Key points

Angle sum of a triangle is 180°; an exterior angle equals the sum of the two interior opposite angles.
Sum of interior angles of an n-sided polygon = (n − 2) × 180°; each exterior angle of a regular polygon = 360°/n.
Centroid divides each median in the ratio 2:1; in an equilateral triangle centroid, incentre, circumcentre and orthocentre coincide.
Angle in a semicircle is 90°; the tangent is perpendicular to the radius at the point of contact.
Perpendicular from the centre bisects a chord.

Explanation

Similar triangles: corresponding sides are proportional; areas are in the ratio of squares of sides.
Common Pythagorean triples: 3-4-5, 5-12-13, 8-15-17, 7-24-25.
Angles in the same segment are equal; opposite angles of a cyclic quadrilateral sum to 180°.
Two tangents from an external point are equal in length.

Step by step

Draw and label.
Pick the theorem.
Compute and check angle sums.

Examples

Angles of a triangle are in the ratio 2:3:4 → 40°, 60°, 80°.
Radius 13 cm, chord at distance 5 cm from the centre → half chord = √(169 − 25) = 12 → chord = 24 cm.
Each exterior angle of a regular polygon is 24° → number of sides = 360/24 = 15.

Traps and advanced points

Trap: Using the interior-angle formula for a single exterior angle.
Trap: Forgetting that only the perpendicular from the centre bisects a chord.
Trap: Taking the area ratio of similar triangles equal to the side ratio.

Quick revision

(n − 2) × 180°.
Centroid 2:1.
Semicircle angle 90°; tangent ⟂ radius.

Practice MCQs

Two angles of a triangle are 32° and 49°. Find the third angle.
1. 99°
2. 109°
3. 104°
4. 94°
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Answer: (a) 99°. Sum of angles in a triangle = 180°.
Two angles of a triangle are 38° and 56°. Find the third angle.
1. 96°
2. 91°
3. 86°
4. 81°
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Answer: (c) 86°. Sum of angles in a triangle = 180°.
Two angles of a triangle are 43° and 71°. Find the third angle.
1. 66°
2. 76°
3. 61°
4. 71°
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Answer: (a) 66°. Sum of angles in a triangle = 180°.
Two angles of a triangle are 49° and 38°. Find the third angle.
1. 98°
2. 88°
3. 93°
4. 103°
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Answer: (c) 93°. Sum of angles in a triangle = 180°.
Two angles of a triangle are 54° and 48°. Find the third angle.
1. 78°
2. 73°
3. 83°
4. 88°
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Answer: (a) 78°. Sum of angles in a triangle = 180°.
Using π=22/7, find the circumference of a circle of radius 112 cm.
1. 704 cm
2. 726 cm
3. 748 cm
4. 682 cm
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Answer: (a) 704 cm. Circumference = 2πr.
Using π=22/7, find the circumference of a circle of radius 476 cm.
1. 2970 cm
2. 3014 cm
3. 2992 cm
4. 3036 cm
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Answer: (c) 2992 cm. Circumference = 2πr.
Two angles of a triangle are 35° and 50°. Find the third angle.
1. 105°
2. 100°
3. 95°
4. 90°
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Answer: (c) 95°. Sum of angles in a triangle = 180°.
Using π=22/7, find the circumference of a circle of radius 441 cm.
1. 2772 cm
2. 2816 cm
3. 2794 cm
4. 2750 cm
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Answer: (a) 2772 cm. Circumference = 2πr.
Two angles of a triangle are 26° and 77°. Find the third angle.
1. 82°
2. 72°
3. 87°
4. 77°
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Answer: (d) 77°. Sum of angles in a triangle = 180°.
Using π=22/7, find the circumference of a circle of radius 406 cm.
1. 2596 cm
2. 2574 cm
3. 2530 cm
4. 2552 cm
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Answer: (d) 2552 cm. Circumference = 2πr.
Two angles of a triangle are 52° and 39°. Find the third angle.
1. 84°
2. 89°
3. 94°
4. 99°
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Answer: (b) 89°. Sum of angles in a triangle = 180°.
Two angles of a triangle are 47° and 79°. Find the third angle.
1. 59°
2. 54°
3. 49°
4. 64°
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Answer: (b) 54°. Sum of angles in a triangle = 180°.
Using π=22/7, find the circumference of a circle of radius 273 cm.
1. 1738 cm
2. 1716 cm
3. 1694 cm
4. 1760 cm
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Answer: (b) 1716 cm. Circumference = 2πr.
Using π=22/7, find the circumference of a circle of radius 266 cm.
1. 1694 cm
2. 1672 cm
3. 1650 cm
4. 1716 cm
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Answer: (b) 1672 cm. Circumference = 2πr.

Geometry

Overview

Key points

Explanation

Step by step

Examples

Traps and advanced points

Quick revision

Practice MCQs

More topics in this chapter