Triangles

Master the concepts of triangles with our comprehensive guide. Learn about different types of triangles, their properties, and applications.

Table of Contents

Simplified Quantitative Formulas: Triangles

  • Sum of Angles: The sum of interior angles of any triangle = 180°.
  • Types by Sides: Scalene (all sides different), Isosceles (two equal), Equilateral (all equal, each angle 60°).
  • Types by Angles: Right-angled (one 90°), Acute (all < 90°), Obtuse (one > 90°).
  • Pythagorean Theorem: a² + b² = c² (c = hypotenuse in right triangle).
  • Area: Area = ½ × base × height or Area = ½ × ab × sin(C)
  • Heron's Formula: For sides a, b, c, s = (a+b+c)/2, Area = √[s(s-a)(s-b)(s-c)]
  • Special Points: Centroid (intersection of medians, divides in 2:1), Incenter (angle bisectors, center of incircle), Circumcenter (perpendicular bisectors, center of circumcircle).
  • Equilateral Triangle: Circumradius R = a/√3, Inradius r = a/(2√3)
  • Similarity & Congruence: Triangles are similar if angles are equal and sides are proportional. Congruent if all sides and angles are equal.
  • Basic Proportionality Theorem (Thales): If a line is drawn parallel to one side of a triangle, it divides the other two sides proportionally.
  • Variable Definitions: a, b, c = side lengths; s = semi-perimeter; h = height; R = circumradius; r = inradius

What do these mean? (Super Simple Explanations & Examples)

  • Sum of Angles: In any triangle, the three angles add up to 180°. Example: If two angles are 50° and 60°, the third is 70°.
  • Pythagorean Theorem: In a right triangle with sides 3, 4, 5: 3²+4²=9+16=25=5².
  • Area: If base = 6, height = 4, area = ½×6×4 = 12.
  • Heron's Formula: Sides 5, 6, 7: s = (5+6+7)/2 = 9. Area = √[9×4×3×2] = √216 ≈ 14.7
  • Special Points: Centroid divides median in 2:1 ratio. Incenter is center of inscribed circle.
  • Equilateral Triangle: Side a = 6, R = 6/√3 ≈ 3.46, r = 6/(2√3) ≈ 1.73
  • Similarity: If two triangles have angles 30°, 60°, 90°, and sides in ratio 1:2:√3, they are similar.
  • Basic Proportionality Theorem: If DE || BC in triangle ABC, then AD/DB = AE/EC.
  • Variable Definitions: a, b, c = side lengths; s = semi-perimeter; h = height; R = circumradius; r = inradius

1. Introduction

Understanding Triangles

A triangle is a polygon with three edges and three vertices. It is one of the basic shapes in geometry.

Basic Elements of a Triangle:

  • Three sides (a, b, c)
  • Three angles (A, B, C)
  • Three vertices
  • Sum of angles = 180°
Basic Triangle
A B C a b c

2. Types of Triangles

Classification by Sides

Equilateral Triangle
A B C a a a

All sides equal

Isosceles Triangle
A B C a b b

Two sides equal

Scalene Triangle
A B C a b c

All sides different

Classification by Angles

Acute Triangle
A B C 60° 60° 60°

All angles < 90°

Right Triangle
A B C 90°

One angle = 90°

Obtuse Triangle
A B C 120°

One angle > 90°

3. Properties

Basic Properties

Angle Properties:

  • Sum of angles = 180°
  • Exterior angle = Sum of opposite interior angles
  • Sum of exterior angles = 360°

Side Properties:

  • Sum of any two sides > third side
  • Difference of any two sides < third side
  • Longest side is opposite to largest angle
Triangle Properties
Exterior Angle A B C α β γ

4. Important Theorems

Key Theorems with Proofs

1. Pythagoras Theorem

a² + b² = c²

(For right-angled triangles)

Proof:

  1. Consider a right-angled triangle ABC with right angle at C
  2. Draw squares on all three sides of the triangle
  3. The area of the square on the hypotenuse (c²) equals the sum of areas of squares on other two sides (a² + b²)
  4. This can be proven using similar triangles and area relationships
Pythagoras Theorem Proof
A B C a b c 90°

2. Angle Sum Theorem

∠A + ∠B + ∠C = 180°

Proof:

  1. Draw a line parallel to one side of the triangle through the opposite vertex
  2. This creates alternate angles that are equal
  3. The angles on the straight line add up to 180°
  4. Therefore, the sum of angles in the triangle is 180°
Angle Sum Theorem Proof
A B C α β

3. Exterior Angle Theorem

Exterior angle = Sum of opposite interior angles

Proof:

  1. Extend one side of the triangle
  2. The exterior angle and the adjacent interior angle form a linear pair (180°)
  3. The sum of all interior angles is 180°
  4. Therefore, the exterior angle equals the sum of opposite interior angles
Exterior Angle Theorem Proof
A B C θ

4. Similarity Theorems

AAA Similarity:

  1. If corresponding angles are equal, triangles are similar
  2. This is the most fundamental similarity criterion
  3. Implies proportional sides

SAS Similarity:

  1. If two sides are proportional and included angle is equal
  2. Triangles are similar
  3. Third side will automatically be proportional

SSS Similarity:

  1. If all three sides are proportional
  2. Triangles are similar
  3. Corresponding angles will be equal
Similar Triangles
A B C D E F

5. Area Theorems

Basic Area Formula:

Area = (1/2) × base × height

  1. Draw a perpendicular from vertex to base
  2. This creates two right triangles
  3. Area of each right triangle = (1/2) × base × height
  4. Total area = Sum of areas of both triangles

Heron's Formula:

Area = √(s(s-a)(s-b)(s-c)) where s = (a+b+c)/2

  1. Calculate semi-perimeter s = (a+b+c)/2
  2. Subtract each side from s
  3. Multiply all results
  4. Take square root of the product
Area Calculation
A B C h b

5. Applications

Real-World Applications

Common Applications:

  • Architecture and construction
  • Navigation and surveying
  • Computer graphics and gaming
  • Engineering and design
Example 1
Find the height of a building using a right-angled triangle if the distance from the building is 20m and the angle of elevation is 30°.
Solution:
1. Using tan(30°) = height/20
2. height = 20 × tan(30°)
3. height = 20 × 0.577
4. height ≈ 11.54m

Practice Questions

Test your understanding of Triangles with 20 fully solved, step-by-step questions designed for beginners.

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