# Solving Right Triangles

A **right triangle** has a **right angle** (90º) and two acute angles.

#### Hypotenuse

The **hypotenuse** is the **side opposite** the **right angle**, and is the largest side of the triangle.

#### Legs or Catheti

The **legs** or **catheti** (singular: cathetus) are the **sides opposite** the **acute angles** and are the shorter sides of the triangle.

### Pythagorean Theorem

In a right triangle, the **square of the hypotenuse equals the sum of the squares of the legs**.

### Area of a Right Triangle

The area of a right triangle is equal to** half the product of the legs**.

## Solving Right Triangles

To solve a triangle, we must find the value of all the sides and angles.

To solve a right triangle, we use trigonometric ratios and the Pythagorean theorem.

### 1. The Hypotenuse and a Leg Are Known

#### Example

Solve a right triangle knowing:

a = 415 m and b = 280 m.

sin B = 280/415 = 0.6747 B = arcsin 0.6747 = 42° 25′

C = 90° - 42° 25′ = 47° 35′

c = a cos B c = 415 · 0.7381 = 306. 31 m

### 2. Two Legs Are Known

#### Example

Solve a right triangle knowing:

b = 33 m and c = 21 m.

tan B = 33/21 = 1.5714 B = 57° 32′

C = 90° - 57° 32′ = 32° 28′

a = b/sin B a = 33/0.8347 = 39.12 m

### 3. The Hypotenuse and an Acute Angle Are Known

#### Example

Solve a right triangle knowing:

a = 45 m y B = 22°.

C = 90° − 22° = 68°

b = a sin 22° b = 45 · 0.3746 = 16.85 m

c = a cos 22° c = 45 · 0.9272 = 41.72 m

### 4. A Leg and an Acute Angle Are Known

#### Example

Solve a right triangle knowing:

b = 5.2 m and B = 37º

C = 90° − 37° = 53º

a = b/sin B a = 5.2/0.6018 = 8.64 m

c = b · cot B c = 5.2 · 1.3270 = 6. 9 m