The Pythagorean Theorem

Learning Objectives

Identify the hypotenuse and the legs of a right triangle.

State the Pythagorean Theorem (a² + b² = c²).

Use the Pythagorean Theorem to find the length of a missing side of a right triangle.

Apply the theorem to solve simple real-world problems.

The Secret of the Right Triangle

Over 2,500 years ago, the Greek mathematician Pythagoras discovered a special relationship between the sides of a right triangle. A right triangle is any triangle that has one angle that is exactly 90 degrees (a 'right angle'). This relationship is one of the most famous and useful theorems in all of mathematics.

The Parts of a Right Triangle

Legs: The two sides of the right triangle that form the right angle. We usually label them a and b.

Hypotenuse: The side opposite the right angle. It is always the longest side of the right triangle. We label it c.

The Pythagorean Theorem

The Pythagorean Theorem states that in any right triangle, the square of the length of the hypotenuse is equal to the sum of the squares of the lengths of the two legs.

As an equation, it is written:

a² + b² = c²

This theorem works for every right triangle, but only for right triangles.

Using the Theorem to Find a Missing Side

We can use this formula to find the length of any side of a right triangle if we know the lengths of the other two sides.

Example 1: Finding the Hypotenuse

A right triangle has legs of length 6 cm and 8 cm. Find the length of the hypotenuse.

1.Write the formula: a² + b² = c²

2.Plug in the known values: `6² + 8² = c²`

3.Solve the squares: `36 + 64 = c²`

4.Add the numbers: `100 = c²`

5.Find the square root: To find c, we take the square root of both sides. `√100 = c`

6.Answer: `c = 10 cm`

Example 2: Finding a Missing Leg

A right triangle has a hypotenuse of 13 inches and one leg of 5 inches. Find the length of the other leg.

1.Write the formula: a² + b² = c²

2.Plug in the known values: `5² + b² = 13²`

3.Solve the squares: `25 + b² = 169`

4.Isolate b²: Subtract 25 from both sides. `b² = 169 - 25`

5.Subtract: `b² = 144`

6.Find the square root: `b = √144`

7.Answer: `b = 12 inches`

The theorem is incredibly useful in real-world applications, such as construction (making sure corners are square), navigation (finding the shortest distance), and design.

Key Terms

**Right Triangle: A triangle with one angle measuring 90 degrees.
**Legs: The two sides of a right triangle that form the right angle.
**Hypotenuse: The side of a right triangle opposite the right angle
**Pythagorean Theorem: The rule stating that for any right triangle, the square of the hypotenuse is equal to the sum of the squares of the two legs (a² + b² = c²).
**Square (of a number): The result of multiplying a number by itself. For example, the square of 5 is 25 (5²=25).
**Square Root: A value that, when multiplied by itself, gives the original number. For example, the square root of 25 is 5 (√25=5).

The Pythagorean Theorem

Learning Objectives

The Secret of the Right Triangle

The Parts of a Right Triangle

The Pythagorean Theorem

Using the Theorem to Find a Missing Side

Key Terms

Check Your Understanding