Geometry Basics – Quantitative Aptitude
(IIT‑JEE / Competitive Exam Focus)
Objective – Build a solid foundation in plane geometry that can be recalled quickly under exam conditions.
Style – Concise, formula‑heavy, with short proofs/derivations, and a “trick” column that flags common pitfalls.
1. Fundamental Definitions
| Concept | Symbol | Key Property |
|---|---|---|
| Point | – | Zero‑dimensional, no size |
| Line | ℓ | Infinite in both directions; defined by two distinct points |
| Plane | Π | Flat, two‑dimensional; defined by three non‑collinear points |
| Collinear | – | All points lie on a single line |
| Parallel | ℓ₁ ∥ ℓ₂ | No intersection; equal slopes |
| Perpendicular | ℓ₁ ⟂ ℓ₂ | Product of slopes = –1 |
| Angle | ∠ABC | Measure between two intersecting lines |
| Congruent | ≅ | Identical in shape & size |
| Similar | ∼ | Same shape, different size (ratios of sides equal) |
2. Basic Shapes & Their Properties
| Shape | Key Features | Formulae (Area, Perimeter) |
|---|---|---|
| Triangle | 3 sides, 3 angles | Perimeter = a+b+c Area (Heron) = √[s(s–a)(s–b)(s–c)], s=(a+b+c)/2 Area (½ × base × height) |
| Right Triangle | One 90° angle | Pythagoras: a²+b²=c² Area = (ab)/2 |
| Isosceles Triangle | Two equal sides | Base angles equal |
| Equilateral Triangle | All sides equal | Area = (√3/4)a² |
| Quadrilateral | 4 sides | Pythagoras for cyclic, Brahmagupta for cyclic |
| Rectangle | Opposite sides equal, all angles 90° | Perimeter = 2(l+w) Area = l × w |
| Square | All sides equal, all angles 90° | Perimeter = 4a Area = a² |
| Parallelogram | Opposite sides parallel & equal | Area = base × height |
| Rhombus | All sides equal | Area = (d₁ × d₂)/2 |
| Trapezium (Trapezoid) | One pair of parallel sides | Area = (a+b)/2 × h |
| Circle | All points equidistant from centre | Circumference = 2πr Area = πr² |
Tip – Always start with “simple” formulas (e.g., ½ × base × height) before moving to Heron’s or Brahmagupta’s.
3. Angle Relationships (Key for Tricky Problems)
| Situation | Relationship | Quick Test |
|---|---|---|
| Adjacent angles on a straight line | Sum = 180° | Check with “∠X + ∠Y = 180°” |
| Angles of a triangle | Sum = 180° | Always |
| Interior angles of a regular polygon with n sides | ((n–2) × 180°)/n | For triangle → 60°, square→90° |
| Angles formed by two intersecting lines | Vertical angles equal | ∠A = ∠C, ∠B = ∠D |
| Angles between a line & a tangent to a circle | Complementary to radius | 90° – angle between radius & tangent |
| Angles formed by two chords intersecting inside a circle | Sum of opposite angles = 180° | ∠A + ∠C = 180° |
| Central vs. Inscribed angles subtended by the same arc | Central = 2 × Inscribed | Useful in circle problems |
Common Pitfall: Confusing opposite angles in a parallelogram (they are equal) with vertical angles (also equal but from different intersection).
4. Coordinate Geometry Basics (for “IIT‑JEE” level)
| Concept | Formula | Example |
|---|---|---|
| Distance between (x₁, y₁) and (x₂, y₂) | √[(x₂–x₁)² + (y₂–y₁)²] | Distance AB |
| Mid‑point | ((x₁+x₂)/2 , (y₁+y₂)/2) | Midpoint of AB |
| Slope of line through (x₁, y₁),(x₂, y₂) | m = (y₂–y₁)/(x₂–x₁) | Slope of AB |
| Equation of line | y–y₁ = m(x–x₁) | Use slope‑intercept |
| Area of triangle with vertices (x₁,y₁),(x₂,y₂),(x₃,y₃) | ½ | x₁(y₂–y₃)+x₂(y₃–y₁)+x₃(y₁–y₂) |
| Circle (center (h,k), radius r) | (x–h)²+(y–k)² = r² | Standard form |
Tricky Point: When slopes are undefined (vertical line), the equation is x = constant.
5. Key Theorems & Their Quick Derivations
| Theorem | Quick Proof Idea | Practical Use |
|---|---|---|
| Pythagoras | In right triangle, square of hypotenuse = sum of squares of legs. | Finding missing side, verifying rightness |
| Thales’ Theorem | If AB is diameter of circle, ∠ACB is right. | Pro |