Geometry
SSC-CGL Exams
1. Overview
Geometry tests your understanding of lines, angles, triangles, circles, and theorems. SSC questions focus on quick formula applications, property-based problem-solving, and diagram interpretation.
2. Lines and Angles
Key Concepts
Straight Line: Angle with horizontal = θ → slope = tan θ
Parallel Lines: Alternate interior angles = equal, Corresponding angles = equal
Perpendicular Lines: Product of slopes = −1
Sum of angles on a line: 180°
Sum of angles at a point: 360°
Example 1
If ∠A + ∠B = 180°, and ∠A = 65°, find ∠B
∠B = 180 − 65 = 115°
115°
Example 2
Two perpendicular lines, slope of one = 3, find slope of other.
m₁ × m₂ = −1 ⇒ 3 × m₂ = −1 ⇒ m₂ = −1/3
−1/3
3. Triangles
Properties
| Concept | Formula / Property |
|---|---|
| Sum of angles | 180° |
| Pythagoras | a² + b² = c² (right triangle) |
| Area | ½ × base × height |
| Heron's formula | A = √[s(s-a)(s-b)(s-c)], s = (a+b+c)/2 |
| Triangle inequality | a + b > c, etc. |
Types of Triangles
Equilateral
All sides equal, all angles 60°
Isosceles
Two sides equal, two angles equal
Right-angled
One 90° angle, use Pythagoras
Scalene
No equal sides or angles
Example 3
Triangle sides 6, 8, 10 → Area?
Right triangle check: 6² + 8² = 36+64=100=10² ✅ right triangle
A = ½ × 6 × 8 = 24 units²
Example 4
Triangle sides 7, 8, 9 → Area using Heron's formula:
s = (7+8+9)/2 = 12
A = √[12(12−7)(12−8)(12−9)] = √[12×5×4×3] = √720 ≈ 26.83
SSC Shortcut:
- Right triangle → use base × height /2
- Equilateral → Area = (√3/4 × a²)
- Heron → Use only for scalene triangles
4. Circles
Properties
| Concept | Formula |
|---|---|
| Circumference | 2πr |
| Area | πr² |
| Angle at centre | ∠ at centre = 2 × ∠ at circumference |
| Tangent | Perpendicular to radius at point of contact |
| Chord | Line segment joining two points on circle |
Theorems
Angle in semicircle = 90°
Tangent-radius theorem: Tangent ⊥ radius
Cyclic quadrilateral: Opposite angles sum = 180°
Example 5
Circle radius = 7 cm → Area & Circumference
A = π7² = 49π, C = 2π7 = 14π
Example 6
Tangent meets radius → angle = ?
90°
5. Theorems & Applications
| Theorem | Practical Use (SSC) |
|---|---|
| Pythagoras | Find missing side, height, diagonal of square/rectangle |
| Triangle sum | Check angles |
| Congruence & similarity | Solve missing sides/angles |
| Circle theorems | Find angles or tangent lengths |
| Quadrilateral sum | Solve interior angle problems |
Example 7
Square diagonal = 10 cm → side?
s² + s² = 10² ⇒ 2s² = 100 ⇒ s² = 50 ⇒ s = √50 ≈ 7.07 cm
Example 8
Two chords of a circle intersect inside → segments a, b, c, d → a×b = c×d
SSC formula: a×b = c×d
Example 9
Right triangle inscribed in circle → hypotenuse = diameter
Hypotenuse = diameter of circle
6. SSC Short Tricks / Tips
Triangle angles sum = 180° → check missing quickly
Pythagoras always useful for rectangles, squares, diagonals
Circle → Tangent ⊥ radius → angles with tangents appear frequently
Chord length formula: l = 2√(r²−d²), d = distance from center
Square/circle → memorize diagonal formulas: square d = a√2, rectangle d = √(l² + b²)
7. Practice Section
Q1. Triangle sides 5, 12, 13 → Area?
View Answer
Right triangle: ½ × 5 × 12 = 30
30
Q2. Square diagonal = 10 → Side?
View Answer
s = 10/√2 ≈ 7.07
7.07
Q3. Circle radius = 14 → Circumference & Area
View Answer
C = 2π14 = 28π, A = π14² = 196π
28π, 196π
Q4. Right triangle inscribed in circle → diameter? Hypotenuse = ?
View Answer
Hypotenuse = diameter of circle
Diameter = hypotenuse
Q5. Tangent touches circle → angle with radius?
View Answer
90°
Q6. Chord segments a=3, b=4 → intersect with chord c=6, find d?
View Answer
a×b = c×d ⇒ 3×4 = 6×d ⇒ d=2
2
8. Quick Recap Table
| Topic | Key Formula / Property | SSC Tip |
|---|---|---|
| Lines | Parallel → alt/corresponding angles equal | Draw diagram |
| Angles | Sum on line =180°, sum at point=360° | Check sums |
| Triangle | Sum angles =180°, Area=½bh, Heron | Identify type quickly |
| Circle | Circumference 2πr, Area πr² | Tangent ⊥ radius |
| Square/Rectangle | Diagonal √(l²+b²) | For diagonals & triangles |
| Chord | a×b = c×d (intersecting chords) | SSC common trick |
| Right triangle in circle | Hypotenuse = diameter | Easy recognition |
You've completed Article 11: Geometry!
Courage Tip: For SSC, focus on diagonals, intersecting chords, tangents, and triangle-circle connections — these are frequently tested.
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