Geometry Basics – Quantitative Aptitude
For the Goa Police Constable Examination
1. Simple Explanation
Geometry in quantitative aptitude deals with the study of shapes, sizes, and relative positions of figures in space.
In the exam you’ll encounter:
- Basic shapes – triangles, rectangles, squares, circles, regular polygons.
- Key properties – sides, angles, symmetry, congruence.
- Formulas – area, perimeter, circumference, volume (if any).
- Coordinate geometry – distance, mid‑point, slope (usually 2‑D).
Mastering these fundamentals lets you solve most geometry‑related questions quickly.
2. Key Points (Cheat‑Sheet)
| Topic | Important Facts | Typical Questions |
|---|---|---|
| Triangles | • 3 sides, 3 angles. • Types: scalene, isosceles, equilateral, right, obtuse, acute. • Sum of interior angles = 180°. |
• Find missing angle. • Verify if a triangle is right. • Compute perimeter/area. |
| Pythagoras | For a right triangle: (a^2 + b^2 = c^2) (c = hypotenuse). | • Find missing side. • Check if a triangle is right. |
| Quadrilaterals | • 4 sides, 4 angles. • Special cases: rectangle (opposite sides equal, angles 90°), square (all sides equal & 90°), rhombus (all sides equal, opposite angles equal). |
• Perimeter of given figure. • Identify shape from side/angle data. |
| Circles | • Radius (r), diameter (d = 2r). • Circumference (C = \pi d = 2\pi r). • Area (A = \pi r^2). • Arc length (L = \frac{\theta}{360°},C). |
• Find radius from area/circumference. • Compute arc length of a sector. |
| Polygons | • Polygon with (n) sides. • Sum of interior angles (= (n-2)\times180°). • For regular polygons, each interior angle (= \frac{(n-2)\times180°}{n}). |
• Find missing angles. • Determine number of sides from interior angle. |
| Coordinate Geometry | • Distance between ((x_1,y_1)) and ((x_2,y_2)): (\displaystyle d=\sqrt{(x_2-x_1)^2+(y_2-y_1)^2}). • Mid‑point: (\displaystyle \big(\frac{x_1+x_2}{2},,\frac{y_1+y_2}{2}\big)). • Slope of line (m=\frac{y_2-y_1}{x_2-x_1}). |
• Find distance between two points. • Verify if a line is perpendicular (product of slopes = –1). |
Tip: For time‑saving, memorize the most common formulas (area of triangle, circle, rectangle, etc.) and the Pythagorean theorem.
3. Worked Example
Question:
A right‑angled triangle has one leg of length 6 cm and a hypotenuse of 10 cm. Find:
- The length of the other leg.
- The area of the triangle.
- The perimeter of the triangle.
Step‑by‑Step Solution
-
Find the missing leg using Pythagoras
[ a^2 + b^2 = c^2 \quad\Rightarrow\quad 6^2 + b^2 = 10^2 ] [ 36 + b^2 = 100 ;\Rightarrow; b^2 = 64 ;\Rightarrow; b = 8 \text{ cm} ] -
Area of a right‑angled triangle
[ A = \frac{1}{2}\times \text{base}\times \text{height} = \frac{1}{2}\times 6 \times 8 = 24 \text{ cm}^2 ] -
Perimeter
[ P = \text{leg}_1 + \text{leg}_2 + \text{hypotenuse} = 6 + 8 + 10 = 24 \text{ cm} ]
Answer:
- Other leg = 8 cm
- Area = 24 cm²
- Perimeter = 24 cm
4. Quick Revision Summary
| Concept | Formula / Rule | Use‑Case |
|---|---|---|
| Triangle angle sum | (\alpha + \beta + \gamma = 180°) | Find missing angle |
| Pythagoras | (a^2+b^2=c^2) | Find side or check right triangle |
| Area – Triangle | (\frac{1}{2}\times \text{base}\times \text{height}) | Compute area |
| Area – Rectangle / Square | (l\times w |