Right Triangle

A right triangle is a triangle with one angle measuring 90°. Right triangles form the foundation of trigonometry and are crucial for understanding the relationship between angles and sides.

Parts of a Right Triangle

• Hypotenuse (c): The side opposite the 90° angle, and the longest side.
• Legs (a and b): The two shorter sides of the triangle.
• Angles: Angle A opposite side a, Angle B opposite side b, and Angle C = 90°.
• Altitude (h): The line from the right angle to the hypotenuse, dividing the triangle into two smaller, similar triangles.

If all three sides are integers, the triangle is called a Pythagorean triangle and the side lengths form a Pythagorean triple (e.g., 3-4-5, 5-12-13, 8-15-17).

Area and Perimeter

The perimeter is the sum of all sides: P = a + b + c

The area of a right triangle can be calculated as:

A = 1/2 × a × b = 1/2 × c × h

Special Right Triangles

30°-60°-90° Triangle

This triangle has angles of 30°, 60°, and 90°. The side lengths follow the ratio 1 : √3 : 2.

• Angles: 30° : 60° : 90°
• Side ratio: 1 : √3 : 2
• Example: If side opposite 60° (b) = 5

a = b / √3 = 5 / √3
c = b × 2 / √3 = 10 / √3

Knowing one side allows you to calculate the other sides easily. These triangles are used in trigonometry for angles that are multiples of π/6.

45°-45°-90° Triangle

Also called an isosceles right triangle, it has two equal sides and the side ratio follows 1 : 1 : √2.

• Angles: 45° : 45° : 90°
• Side ratio: 1 : 1 : √2
• Example: If hypotenuse (c) = 5

a = c / √2 = 5 / √2

Knowing one side allows calculation of the remaining sides. These triangles are used to evaluate trigonometric functions for multiples of π/4.

Pythagorean Theorem

For any right triangle, the Pythagorean theorem holds:

c² = a² + b²

Key Takeaways

• Right triangles have one 90° angle.
• The hypotenuse is always the longest side.
• Area = 1/2 × (product of the legs) or 1/2 × hypotenuse × altitude.
• Special triangles (30-60-90, 45-45-90) have consistent side ratios for easy calculation.
• Pythagorean triangles have integer side lengths forming Pythagorean triples.