Geometry Basics – Meghalaya PSC (Quantitative Aptitude)
Geometry is the branch of mathematics that deals with shapes, sizes, relative positions, and properties of space.
In PSC exams you’ll mostly see problems on angles, triangles, quadrilaterals, circles, perimeters, and areas.
The following notes give a quick, clear, and exam‑ready overview.
1. Basic Concepts
| Concept | Definition | Quick Tip |
|---|---|---|
| Point | Exact location in space, no size | Denoted by a capital letter (A, B, C…) |
| Line | Collection of points extending infinitely in both directions | Infinite length, no thickness |
| Plane | Flat, two‑dimensional surface | Infinite in all directions |
| Line Segment | Part of a line bounded by two endpoints | Length can be measured |
| Ray | Part of a line that starts at an endpoint and extends infinitely | One endpoint, one direction |
| Right Angle | 90° | Symbol: ∠ = 90° or ⊥ |
2. Angles
| Type | Definition | Properties |
|---|---|---|
| Acute | < 90° | All acute angles are < 90° |
| Right | = 90° | ∠ = 90° |
| Obtuse | > 90° and < 180° | 180° – (sum of two acute) |
| Straight | = 180° | Two opposite rays |
| Complementary | Sum = 90° | ∠A + ∠B = 90° |
| Supplementary | Sum = 180° | ∠A + ∠B = 180° |
Angle Sum Properties
| Figure | Sum of interior angles |
|---|---|
| Triangle | 180° |
| Quadrilateral | 360° |
| Polygon with n sides | (n – 2) × 180° |
3. Triangles
| Type | Definition | Key Properties |
|---|---|---|
| Equilateral | All sides equal | All angles 60° |
| Isosceles | Two sides equal | Base angles equal |
| Scalene | All sides different | All angles different |
| Right | One right angle | Hypotenuse opposite 90° |
| Obtuse | One obtuse angle | Only one obtuse angle |
| Acute | All angles < 90° | None is right or obtuse |
Important Theorems
| Theorem | Statement |
|---|---|
| Pythagoras | In a right triangle, (a^2 + b^2 = c^2) (c = hypotenuse) |
| Angle Bisector | Divides opposite side in ratio of adjacent sides |
| Median | Connects vertex to midpoint of opposite side |
| Altitude | Perpendicular from vertex to base |
Area Formulas
| Figure | Formula |
|---|---|
| Triangle | (\displaystyle \frac{1}{2} \times \text{base} \times \text{height}) |
| Right Triangle | (\displaystyle \frac{1}{2} \times a \times b) |
| General (Heron) | (\displaystyle \sqrt{s(s-a)(s-b)(s-c)}) where (s=\frac{a+b+c}{2}) |
4. Quadrilaterals
| Shape | Key Properties | Area (if needed) |
|---|---|---|
| Parallelogram | Opposite sides parallel & equal | ( \text{base} \times \text{height}) |
| Rectangle | All angles 90° | ( \text{length} \times \text{breadth}) |
| Rhombus | All sides equal | ( \frac{d_1 \times d_2}{2}) |
| Square | Rectangle + Rhombus | ( \text{side}^2) |
| Trapezium (US)/Trapezoid (UK) | One pair of parallel sides | ( \frac{(a+b)}{2} \times \text{height}) |
5. Circles
| Term | Definition |
|---|---|
| Circumference | Total distance around the circle |
| Radius | Distance from centre to any point on circle |
| Diameter | 2 × Radius |
| Chord | Line segment whose endpoints lie on circle |
| Arc | Part of circumference |
| Sector | Region bounded by two radii and an arc |
| Sector Angle | Angle at centre subtending the arc |
Key Formulas
| Quantity | Formula |
|---|---|
| Circumference | (C = 2\pi r) |
| Area | (A = \pi r^2) |
| Length of Arc (θ in degrees) | (L = \frac{\theta}{360} \times 2\pi r) |
| Length of Arc (θ in radians) | (L = r\theta) |
| Area of Sector (θ in degrees) | (A = \frac{\theta}{360} \times \pi r^2) |
| Area of Sector (θ in radians) | (A = \frac{1}{2} r^2 \theta) |
6. Perimeter & Area – Quick Reference
| Figure | Perimeter | Area |
|---|---|---|
| Rectangle | (2(l + b)) | (l \times b) |
| Square | (4a) | (a^2) |
| Triangle | (a + b + c) | (\frac{1}{2}\ |