Geometry Basics – Study Notes
Program: Assam Patwari / Lekhpal
Subject: Quantitative Aptitude
Topic: Geometry Basics
Goal: Master the core geometric concepts and formulas that appear in the Patwari / Lekhpal exam.
1. Simple Explanation
Geometry is the branch of mathematics that deals with shapes, sizes, and the properties of space. In the Patwari / Lekhpal exam you’ll mainly encounter:
- Basic shapes (points, lines, planes, angles, triangles, quadrilaterals, circles).
- Measurements (perimeter, area, circumference).
- Relationships between different elements (e.g., opposite angles, complementary angles).
Knowing these fundamentals lets you solve problems about land measurements, distance, and simple construction tasks.
2. Key Points
| Concept | What It Is | Key Formula(s) | Typical Use in Exam |
|---|---|---|---|
| Point | Exact location in space | – | Base for defining lines and planes |
| Line | Straight path with no thickness, infinite in both directions | – | Defines angles, segments |
| Line Segment | Part of a line between two points | Length (= | AB |
| Angle | Formed by two rays sharing a vertex | Types: acute, right, obtuse, straight, reflex | Classification, angle addition/subtraction |
| Triangle | 3-sided polygon | Perimeter (P = a+b+c) Area (base × height ÷ 2) Heron’s formula |
Land plots, triangular sections |
| Quadrilateral | 4-sided polygon | Perimeter (P = a+b+c+d) Area (rectangular: (l \times w); parallelogram: base × height) |
Rectangular fields, building footprints |
| Circle | All points at a fixed distance from center | Radius (r), diameter (D = 2r) Circumference (C = 2\pi r) Area (A = \pi r^2) |
Circular ponds, lamp posts |
| Parallel Lines | Never intersect | If a transversal cuts them, alternate interior angles are equal | Checking land boundaries |
| Perpendicular Lines | Intersect at 90° | Right angles | Construction, building layouts |
| Complementary & Supplementary Angles | Sum to 90° or 180° | (x + y = 90^\circ) or (180^\circ) | Quick angle calculations |
Quick Reference Tables
1. Triangles
| Triangle Type | Key Property |
|---|---|
| Equilateral | All sides and angles equal (60° each) |
| Isosceles | Two sides equal; base angles equal |
| Scalene | All sides and angles different |
| Right | One angle = 90° (Pythagoras: (a^2 + b^2 = c^2)) |
2. Quadrilaterals
| Shape | Key Property |
|---|---|
| Rectangle | Opposite sides equal, 4 right angles |
| Square | All sides equal, 4 right angles |
| Parallelogram | Opposite sides equal, opposite angles equal |
| Trapezium (US)/Trapezoid (UK) | One pair of parallel sides |
| Rhombus | All sides equal, opposite angles equal |
3. Circle
| Feature | Formula |
|---|---|
| Circumference | (C = 2\pi r) |
| Area | (A = \pi r^2) |
| Arc length | (L = \frac{\theta}{360^\circ} \times C) (θ in degrees) |
3. Worked Example
Problem (Typical Patwari Question)
*A rectangular plot of land measures 120 m in length and 75 m in width. The plot is divided into 3 equal sections along its length. Find:
- The area of the whole plot.
- The area of one section.
- The perimeter of one section.*
Step‑by‑Step Solution
-
Area of the whole plot
[ A_{\text{whole}} = \text{Length} \times \text{Width} = 120 \text{ m} \times 75 \text{ m} = 9{,}000 \text{ m}^2 ] -
Area of one section
The plot is divided into 3 equal parts along the length, so each section’s length = (120/3 = 40) m.
[ A_{\text{section}} = 40 \text{ m} \times 75 \text{ m} = 3{,}000 \text{ m}^2 ] -
Perimeter of one section
Perimeter = (2(\text{length} + \text{width})).
[ P_{\text{section}} = 2(40 + 75) = 2(115) = 230 \text{ m} ]
Answer
- Whole plot area: 9 000 m²
- One section area: 3 000 m²
- One section perimeter: 230 m
4. Quick Revision Summary
-
Angles
- Complementary: (x + y = 90^\circ)
- Supplementary: (x + y = 180^\circ)
-
Triangles
- Perimeter: sum of all sides.
- Area: (\frac{1}{2}\times\text{base}\times\text{height}).
- Pythagoras for right triangles.
-
Quadrilaterals
- Rectangle area: (l \times w).