Key Points

✓The Pythagorean Theorem applies only to right triangles.
✓The formula is a² + b² = c², where 'a' and 'b' are the legs and 'c' is the hypotenuse.
✓The hypotenuse is always the longest side and is opposite the right angle.
✓You can use the theorem to find a missing side length if you know the other two.
✓The theorem has many real-world applications, such as construction and navigation.

The Pythagorean Theorem is a fundamental concept in geometry that describes the relationship between the sides of a right triangle. A right triangle is a triangle that has one angle equal to 90 degrees (a right angle). The side opposite the right angle is called the hypotenuse (c), and the other two sides are called legs (a and b). The theorem states that the sum of the squares of the legs is equal to the square of the hypotenuse: a² + b² = c². This theorem allows you to find the length of any side of a right triangle if you know the lengths of the other two sides. It's a powerful tool used in many areas of math, science, and engineering. Remember, it only applies to right triangles!

Worked Example

A right triangle has legs of length 3 and 4. Find the length of the hypotenuse.

Step 1: Identify the legs (a and b) and the hypotenuse (c). Here, a = 3 and b = 4. We want to find c.
Step 2: Apply the Pythagorean Theorem: a² + b² = c²
Step 3: Substitute the values: 3² + 4² = c²
Step 4: Calculate the squares: 9 + 16 = c²
Step 5: Add the numbers: 25 = c²
Step 6: Take the square root of both sides: √25 = √c²
Step 7: Simplify: 5 = c

Answer: The length of the hypotenuse is 5.

Try It Yourself

1. A right triangle has legs of length 5 and 12. What is the length of the hypotenuse?

2. The hypotenuse of a right triangle is 10, and one leg is 6. Find the length of the other leg.

3. A ladder 17 feet long leans against a wall. The base of the ladder is 8 feet from the wall. How high up the wall does the ladder reach?

Watch Out For These Mistakes

Forgetting to square the side lengths before adding them.
Incorrectly identifying the hypotenuse.
Applying the theorem to non-right triangles.