4.2 Geometry & Trigonometry Basics
Key Takeaways
- Memorise the standard trig table for 0°, 30°, 45°, 60°, 90° — it answers most RRB trig questions in under 20 seconds.
- Angle of elevation/depression problems reduce to right triangles: label opposite, adjacent, hypotenuse, then pick sin, cos, or tan.
- Pythagoras (a^2 + b^2 = c^2) and the Pythagorean triplets (3-4-5, 5-12-13, 8-15-17, 7-24-25) solve most length/height questions without computation.
- The angle between clock hands, the sum of triangle angles to 180°, and the tangent-radius 90° fact are the three geometry constants RRB reuses.
- In similar triangles, ratios of corresponding sides, medians, altitudes, and perimeters are equal; ratios of areas equal the square of the side ratio.
Lines and Angles
An angle is formed where two rays meet. Key types: acute (< 90°), right (= 90°), obtuse (90°-180°), straight (= 180°), reflex (> 180°).
Two angles are complementary if they sum to 90° and supplementary if they sum to 180°.
When a transversal cuts two parallel lines:
- Corresponding angles are equal.
- Alternate interior angles are equal.
- Co-interior (same-side interior) angles are supplementary.
- Vertically opposite angles are always equal, parallel lines or not.
Triangles
Properties you must remember:
- Sum of interior angles = 180°.
- Exterior angle = sum of the two opposite interior angles.
- Sum of any two sides is greater than the third (triangle inequality).
- Area = (1/2) × base × height = (1/2) × a × b × sin C = √(s(s-a)(s-b)(s-c)) where s = (a+b+c)/2 (Heron's formula).
Pythagoras for a right triangle with legs a, b and hypotenuse c: a^2 + b^2 = c^2. Common triplets: (3,4,5), (5,12,13), (8,15,17), (7,24,25), and their multiples.
Similarity — triangles are similar if AAA (or AA), SSS in the same ratio, or SAS with proportional sides and equal included angle. In similar triangles, ratios of corresponding sides, medians, altitudes, and perimeters are equal; ratios of areas equal the square of the ratio of sides.
Congruence rules: SSS, SAS, ASA, AAS, RHS (right angle, hypotenuse, side).
Quadrilaterals
| Shape | Area | Perimeter | Key Property |
|---|---|---|---|
| Square | a^2 | 4a | Diagonals equal, bisect at 90° |
| Rectangle | l × b | 2(l + b) | Diagonals equal |
| Parallelogram | base × height | 2(sum of adjacent sides) | Opposite sides equal and parallel |
| Rhombus | (d1 × d2)/2 | 4 × side | Diagonals bisect at 90° |
| Trapezium | (1/2)(a + b) × h | sum of sides | One pair of parallel sides |
Circles
- Radius r, diameter d = 2r.
- Circumference = 2πr = πd.
- Area = πr^2.
- A tangent is perpendicular to the radius at the point of contact (90° fact).
- A chord perpendicular to a radius is bisected by it.
- Angle at the centre is twice the angle at the circumference subtended by the same arc.
- Length of arc = (θ/360) × 2πr; area of sector = (θ/360) × πr^2.
Coordinate Geometry Basics
For two points (x1, y1) and (x2, y2):
- Distance = √((x2 - x1)^2 + (y2 - y1)^2).
- Midpoint = ((x1 + x2)/2, (y1 + y2)/2).
- Slope m = (y2 - y1)/(x2 - x1).
- Section formula: point dividing in ratio m:n internally is ((mx2 + nx1)/(m+n), (my2 + ny1)/(m+n)).
Trigonometric Ratios
In a right triangle with angle θ:
- sin θ = opposite / hypotenuse
- cos θ = adjacent / hypotenuse
- tan θ = opposite / adjacent
- Reciprocals: cosec θ = 1/sin θ, sec θ = 1/cos θ, cot θ = 1/tan θ.
A reliable mnemonic: SOH-CAH-TOA.
Standard Angles Table
| θ | 0° | 30° | 45° | 60° | 90° |
|---|---|---|---|---|---|
| sin | 0 | 1/2 | 1/√2 | √3/2 | 1 |
| cos | 1 | √3/2 | 1/√2 | 1/2 | 0 |
| tan | 0 | 1/√3 | 1 | √3 | undefined |
Values for 0°, 30°, 45°, 60°, 90° rise for sin (0 to 1) and fall for cos (1 to 0); tan follows sin/cos.
Core Identities
- sin^2 θ + cos^2 θ = 1
- 1 + tan^2 θ = sec^2 θ
- 1 + cot^2 θ = cosec^2 θ
- sin(90° - θ) = cos θ; cos(90° - θ) = sin θ; tan(90° - θ) = cot θ.
Angles of Elevation and Depression
The angle of elevation is measured upward from the horizontal to the line of sight. The angle of depression is measured downward from the horizontal to the line of sight. Both create a right triangle where a height, distance, and angle are linked by a trig ratio.
Worked example 1 (Pythagoras). A ladder 13 m long leans against a wall with its base 5 m from the wall. How high does it reach?
Wall height h, base 5 m, ladder 13 m form a right triangle with hypotenuse 13. So h^2 + 5^2 = 13^2 → h^2 = 169 - 25 = 144 → h = 12 m. This is the (5, 12, 13) triplet — recognise it and skip the arithmetic.
Worked example 2 (elevation). From a point on the ground 30 m away from the foot of a tower, the angle of elevation to the top of the tower is 30°. Find the tower's height.
Right triangle with adjacent = 30 m, opposite = h, angle = 30°. Use tan: tan 30° = h / 30 → 1/√3 = h/30 → h = 30/√3 = 10√3 m ≈ 17.32 m.
Worked example 3 (similar triangles). Two similar triangles have corresponding sides in ratio 3:5. The area of the smaller is 36 cm². Find the area of the larger.
Area ratio = (side ratio)^2 = 9/25. So 36 / A_large = 9/25 → A_large = 36 × 25 / 9 = 100 cm².
Exam Traps
- Confusing angle of elevation with the angle inside the triangle: elevation is measured from the horizontal, not from the vertical.
- Using degrees and radians interchangeably: RRB uses degrees only.
- Forgetting that tan 90° is undefined: if a question's "answer" seems to need tan 90°, re-read — you likely chose the wrong ratio.
- Mixing up sin and cos of complementary angles: sin(90° - θ) = cos θ, not sin θ.
- Approximating √3 as 1.7 and √2 as 1.41 is acceptable, but match the answer choice's exact form (e.g., 10√3) before converting to decimals.
A vertical pole 6 m tall casts a shadow 2√3 m long on level ground. What is the angle of elevation of the sun?
The lengths of the two diagonals of a rhombus are 10 cm and 24 cm. What is the side length of the rhombus?
Two similar triangles have corresponding sides in the ratio 4:9. If the area of the larger triangle is 81 cm², what is the area of the smaller triangle?