Geometry Basics – Quantitative Aptitude
(For Himachal Pradesh Patwari / Lekhpal Examination)
Goal
Master the fundamental concepts, formulas and quick problem‑solving techniques for geometry that are frequently asked in the Patwari / Lekhpal exam.
1. Simple Explanation
Geometry is the branch of mathematics dealing with points, lines, angles, shapes, and their properties. In the Patwari exam you’ll encounter:
| Concept | What you need to know |
|---|---|
| Point, Line, Plane | Basic elements, no dimensions |
| Angle | Types (acute, right, obtuse, straight, reflex) |
| Triangles | Classification, properties, area, perimeter |
| Quadrilaterals | Types, properties, area formulas |
| Circles | Radius, diameter, circumference, area |
| Coordinate Geometry | Distance, mid‑point, slope (optional) |
2. Key Points & Formulas
Tip: Memorise the “Area = ½ × base × height” as it appears in triangles, trapeziums, etc.
2.1 Basic Elements
| Symbol | Meaning | Example |
|---|---|---|
P |
Point | A point on a line |
AB |
Line segment | Between points A and B |
AB (with three letters) |
Ray | AB through point B |
ABC |
Angle at B | ∠ABC |
2.2 Angles
| Type | Measure | Example |
|---|---|---|
| Acute | < 90° | 30°, 45° |
| Right | = 90° | 90° |
| Obtuse | > 90° and < 180° | 120° |
| Straight | = 180° | 180° |
| Reflex | > 180° | 270° |
2.3 Triangles
| Property | Formula | Notes |
|---|---|---|
| Area | (\displaystyle \frac{1}{2} \times \text{base} \times \text{height}) | Height is perpendicular to base |
| Perimeter | (a + b + c) | Sum of all sides |
| Pythagoras (right triangle) | (a^2 + b^2 = c^2) | (c) = hypotenuse |
| Heron’s Formula | (\displaystyle \sqrt{p(p-a)(p-b)(p-c)}) | (p = \frac{a+b+c}{2}) |
| Types | - Equilateral (all sides & angles 60°) - Isosceles (two sides equal) - Scalene (no equal sides) |
2.4 Quadrilaterals
| Shape | Properties | Area Formula |
|---|---|---|
| Rectangle | Opposite sides equal, all angles 90° | (\text{length} \times \text{breadth}) |
| Square | All sides equal, all angles 90° | (s^2) |
| Parallelogram | Opposite sides equal & parallel | (\text{base} \times \text{height}) |
| Trapezium (US) / Trapezoid (UK) | One pair of parallel sides | (\displaystyle \frac{1}{2} \times (\text{sum of parallel sides}) \times \text{height}) |
| Rhombus | All sides equal, diagonals perpendicular | (\displaystyle \frac{d_1 \times d_2}{2}) |
| Kite | Two distinct pairs of adjacent equal sides | (\displaystyle \frac{d_1 \times d_2}{2}) |
2.5 Circles
| Concept | Formula | Notes |
|---|---|---|
| Radius (r) | – | Distance from center to any point on circle |
| Diameter (D) | (D = 2r) | |
| Circumference | (C = 2\pi r) | (\pi \approx 3.1416) |
| Area | (A = \pi r^2) |
2.6 Coordinate Geometry (if asked)
| Formula | Description |
|---|---|
| Distance between ((x_1, y_1)) and ((x_2, y_2)) | (\sqrt{(x_2-x_1)^2 + (y_2-y_1)^2}) |
| Mid‑point | (\left(\frac{x_1+x_2}{2}, \frac{y_1+y_2}{2}\right)) |
| Slope of line through ((x_1, y_1)) and ((x_2, y_2)) | (\frac{y_2-y_1}{x_2-x_1}) |
3. Worked Example
Problem (Patwari‑style):
A rectangular plot has a length of 50 m and a breadth of 30 m. A straight road of width 5 m cuts across the plot from one side to the opposite side, forming two right‑angled triangles on either side of the road. Find the area of the plot that is not covered by the road.
Step‑by‑step solution
-
Plot the situation
- Rectangle: (L = 50) m, (B = 30) m.
- Road width: 5 m (assume it runs parallel to the breadth).
- Two triangles are formed on the sides of the road.
-
Area of whole rectangle
[ A_{\text{rect}} = L \times B = 50 \times 30 = 1500 \text{ m}^2 ] -
Area covered by the road
- Road is a rectangle itself: length (L = 50) m, width (w = 5) m.
[ A_{\text{road}} = 50 \times
- Road is a rectangle itself: length (L = 50) m, width (w = 5) m.