Geometry

🟢 Lite — Quick Review (1h–1d)

Rapid summary for last-minute revision before your exam.

Geometry — Key Facts for GAT Pakistan

Triangle Properties:

Type	Sides	Angles
Equilateral	All equal	All 60°
Isosceles	Two equal	Base angles equal
Scalene	All different	All different
Right-angled	Pythagoras: a² + b² = c²	One 90°

Area Formulas:

Shape	Formula
Triangle	(1/2) × base × height
Rectangle	length × breadth
Square	side²
Circle	πr²
Parallelogram	base × height
Trapezium	(1/2)(a + b) × h

Circle:

• Circumference = 2πr
• Area = πr²
• Area of sector = (θ/360) × πr²

⚡ GAT Exam Tip: π = 22/7 or 3.1416. For quick estimates, use 3.14!

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🟡 Standard — Regular Study (2d–2mo)

Standard content for students with a few days to months.

Geometry — Detailed Study Guide

Angles and Lines

Angle Types:

Type	Measure
Acute	0° < angle < 90°
Right	90°
Obtuse	90° < angle < 180°
Straight	180°
Reflex	180° < angle < 360°

Parallel Lines (Transversal):

• Corresponding angles equal
• Alternate interior angles equal
• Co-interior angles supplementary (180°)

Worked Examples:

Example 1: In a triangle, angles are in ratio 2:3:4. Find all angles.

Solution:
2x + 3x + 4x = 180°
9x = 180°
x = 20°
Angles: 40°, 60°, 80°

Example 2: If two angles of a triangle are 50° and 60°, find third angle.

Solution:
Third angle = 180° - 50° - 60° = 70°

Pythagorean Theorem

For right-angled triangles: $$a^2 + b^2 = c^2 \text{ (where c is hypotenuse)}$$

Pythagorean Triples (remember these):

Triple	Ratio
3:4:5	6:8:10, 9:12:15, 12:16:20
5:12:13	10:24:26
7:24:25	14:48:50
8:15:17	16:30:34

⚡ GAT PYQ: “A ladder 25 m long leans against a wall. If foot is 7m from wall, how far up the wall does it reach?” → Answer: 24 m (7-24-25 triple)

Circle Theorems

Key Theorems:

Theorem	Statement
Angle at center	Central angle = 2 × angle at circumference
Thales	Angle in semicircle = 90°
Equal chords	Equal chords subtend equal angles at center
Perpendicular	Perpendicular from center bisects chord

Tangent Properties:

• Tangent is perpendicular to radius at point of contact
• Two tangents from same external point are equal

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🔴 Extended — Deep Study (3mo+)

Comprehensive coverage for students on a longer study timeline.

Geometry — Complete Notes for GAT

Similarity and Congruency

Congruent Triangles (SSS, SAS, ASA, AAS, RHS):

Rule	Condition
SSS	All three sides equal
SAS	Two sides and included angle equal
ASA	Two angles and included side equal
AAS	Two angles and one side equal
RHS	Right angle, hypotenuse, one side (for right triangles)

Similar Triangles (AAA, AA, SAS):

• Same shape, different sizes
• Corresponding angles equal
• Sides in proportion

Quadrilaterals

Properties:

Quadrilateral	Properties
Parallelogram	Opposite sides parallel, opposite angles equal
Rectangle	Parallelogram with right angles, diagonals equal
Rhombus	Parallelogram with all sides equal, diagonals perpendicular
Square	Rectangle + Rhombus properties
Trapezium	One pair of parallel sides
Kite	Two pairs of adjacent sides equal

Parallelogram Area: Base × Height Rhombus Area: (d₁ × d₂)/2 (where d = diagonals)

3D Geometry

Cuboid:

Property	Formula
Volume	l × b × h
Surface Area	2(lb + bh + hl)
Diagonal	√(l² + b² + h²)

Cube (side = a):

Property	Formula
Volume	a³
Surface Area	6a²
Diagonal	a√3

Cylinder:

Property	Formula
Volume	πr²h
Curved Surface Area	2πrh
Total Surface Area	2πr(r + h)

Cone:

Property	Formula
Volume	(1/3)πr²h
Slant Height	√(r² + h²)
Curved Surface	πrl

Sphere:

Property	Formula
Volume	(4/3)πr³
Surface Area	4πr²

GAT-Style Practice Questions

1. The area of a circle is 154 cm². Find its radius (use π = 22/7).
   (a) 7 cm (b) 14 cm (c) 21 cm (d) 28 cm

   Answer: (a) 7 cm
   Solution: πr² = 154
             (22/7)r² = 154
             r² = 154 × 7/22 = 49
             r = 7 cm

2. A triangle has sides 5, 12, 13. What type of triangle is it?
   (a) Acute (b) Right-angled (c) Obtuse (d) Equilateral

   Answer: (b) Right-angled
   Solution: 5² + 12² = 25 + 144 = 169 = 13²
             Pythagorean triple (5, 12, 13)

3. Find the area of a triangle with base 8 cm and height 5 cm.
   (a) 20 cm² (b) 40 cm² (c) 13 cm² (d) 10 cm²

   Answer: (a) 20 cm²
   Solution: Area = (1/2) × 8 × 5 = 20 cm²

4. The volume of a cylinder is 3080 cm³ and height is 20 cm.
   Find radius (π = 22/7).
   (a) 7 cm (b) 14 cm (c) 21 cm (d) 35 cm

   Answer: (a) 7 cm
   Solution: V = πr²h
             3080 = (22/7) × r² × 20
             r² = 3080 × 7/(22 × 20) = 49
             r = 7 cm

5. In a triangle ABC, angle A = 50°, angle B = 60°. Find angle C.
   (a) 60° (b) 70° (c) 80° (d) 90°

   Answer: (b) 70°
   Solution: A + B + C = 180°
             50° + 60° + C = 180°
             C = 70°

⚡ GAT Strategy: For geometry problems, draw a diagram. Many students lose marks by trying to solve problems mentally!

Coordinate Geometry

Distance Formula: $$d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}$$

Midpoint: $$M = \left(\frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2}\right)$$

Slope of Line: $$m = \frac{y_2 - y_1}{x_2 - x_1}$$

Equation of Line:

• Point-slope: y - y₁ = m(x - x₁)
• Slope-intercept: y = mx + c

Example: Find distance between (3, 4) and (7, 7)

d = √[(7-3)² + (7-4)²]
  = √[16 + 9]
  = √25 = 5

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