Module 3 · SSC CGL Quantitative Aptitude

Geometry & Trigonometry

Lines, angles, triangles, circles, trig ratios.

Geometry Basics · Trigonometry · Height & Distance

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Learning Objectives

Understand the fundamental concepts of Geometry & Trigonometry
Apply key formulas and techniques to solve problems
Practice with exam-level questions to build speed and accuracy

Key Concepts

What is Geometry & Trigonometry?

Geometry deals with the properties and relationships of points, lines, angles, and shapes. Trigonometry deals with the relationships between angles and sides of triangles. SSC CGL Tier 1 includes basic geometry concepts (lines, angles, triangles, circles) and fundamental trigonometric ratios.

Key Concepts (Geometry)

1. Lines & Angles: Vertically opposite angles are equal. Corresponding angles are equal when lines are parallel. Sum of interior angles of a triangle = 180 degrees. Exterior angle = sum of opposite interior angles.

2. Triangle Properties: Pythagorean theorem: a^2 + b^2 = c^2 for right triangles. In an isosceles triangle, angles opposite equal sides are equal. In an equilateral triangle, all angles = 60 degrees. Centroid divides medians in the ratio 2:1.

3. Circles: Angle subtended by a diameter is 90 degrees (Thales theorem). Angles in the same segment are equal. The tangent at any point is perpendicular to the radius at that point.

Key Concepts (Trigonometry)

1. Basic Ratios: sin theta = opposite/hypotenuse. cos theta = adjacent/hypotenuse. tan theta = opposite/adjacent. cosec theta = 1/sin theta. sec theta = 1/cos theta. cot theta = 1/tan theta.

2. Standard Angles: sin 0 = 0, sin 30 = 1/2, sin 45 = 1/sqrt2, sin 60 = sqrt3/2, sin 90 = 1. cos values are reverse of sin. tan 45 = 1.

3. Identities: sin^2 theta + cos^2 theta = 1. sec^2 theta - tan^2 theta = 1. cosec^2 theta - cot^2 theta = 1.

Solved Examples

Example 1 (Geometry — Triangle)

In a right triangle ABC with right angle at B, if AB = 6 and BC = 8, find AC.

a) 9 b) 10 c) 12 d) 14

Solution: b) 10. By Pythagoras: AC^2 = AB^2 + BC^2 = 36 + 64 = 100. AC = sqrt100 = 10.

Example 2 (Trigonometry — Value)

Find the value of sin 60 cos 30 + cos 60 sin 30.

a) 0 b) 1/2 c) 1 d) sqrt3/2

Solution: c) 1. sin 60 = sqrt3/2, cos 30 = sqrt3/2, cos 60 = 1/2, sin 30 = 1/2. Expression = (sqrt3/2)*(sqrt3/2) + (1/2)*(1/2) = 3/4 + 1/4 = 1.

Example 3 (Geometry — Circle)

The angle subtended by a chord at the centre is 120 degrees. What is the angle subtended by it at the circumference?

a) 30 degrees b) 45 degrees c) 60 degrees d) 90 degrees

Solution: c) 60 degrees. The angle at the centre is twice the angle at the circumference. So angle at circumference = 120/2 = 60 degrees.

Shortcut Techniques

For geometry, remember common Pythagorean triplets: 3-4-5, 5-12-13, 7-24-25, 8-15-17, 9-40-41.
For trigonometry, use the mnemonic SOH CAH TOA to recall sin=opp/hyp, cos=adj/hyp, tan=opp/adj.
For complementary angles: sin(90 - theta) = cos theta, tan(90 - theta) = cot theta.
When facing complex trigonometric expressions, convert everything to sin and cos first, then simplify.

Pro Tip: SSC CGL often asks height & distance problems combining geometry and trigonometry. Remember that tan theta = height/distance. For maximum-minimum problems, use the AM-GM inequality or derivatives for optimization.

Advanced Applications

Modern exams feature questions combining multiple quantitative concepts. Master these by practicing previous year papers.

Shortcut Methods

Percentages: Use fraction table (12.5%=1/8, 20%=1/5, 25%=1/4). SI/CI: For 2 years, CI-SI = P(r/100)^2. Time Work: Use LCM of time periods. Average Speed: 2ab/(a+b) for equal distances.

Pro Tip

Write key formulas on rough sheet immediately after the exam bell while memory is fresh.

Advanced Applications

Modern exams feature questions combining multiple quantitative concepts. Master these by practicing previous year papers.

Shortcut Methods

Percentages: Use fraction table (12.5%=1/8, 20%=1/5, 25%=1/4). SI/CI: For 2 years, CI-SI = P(r/100)^2. Time Work: Use LCM of time periods. Average Speed: 2ab/(a+b) for equal distances.

Pro Tip

Write key formulas on rough sheet immediately after the exam bell while memory is fresh.

Advanced Applications

Modern exams feature questions combining multiple quantitative concepts. Master these by practicing previous year papers.

Shortcut Methods

Percentages: Use fraction table (12.5%=1/8, 20%=1/5, 25%=1/4). SI/CI: For 2 years, CI-SI = P(r/100)^2. Time Work: Use LCM of time periods. Average Speed: 2ab/(a+b) for equal distances.

Pro Tip

Write key formulas on rough sheet immediately after the exam bell while memory is fresh.

Advanced Applications

Modern exams feature questions combining multiple quantitative concepts. Master these by practicing previous year papers.

Shortcut Methods

Percentages: Use fraction table (12.5%=1/8, 20%=1/5, 25%=1/4). SI/CI: For 2 years, CI-SI = P(r/100)^2. Time Work: Use LCM of time periods. Average Speed: 2ab/(a+b) for equal distances.

Pro Tip

Write key formulas on rough sheet immediately after the exam bell while memory is fresh.

Right triangle with labeled sides and trigonometric ratios (sin Î¸ = opposite/hypotenuse, cos Î¸ = adjacent/hypotenuse, tan Î¸ = opposite/adjacent)

Practice Questions

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