What is the Pythagorean Theorem?

The Pythagorean Theorem is a fundamental principle in mathematics that deals with right triangles. It relates to the three sides of a right triangle, specifically the two shorter sides known as the legs, and the longest side opposite the right angle, known as the hypotenuse. The theorem states that the square of the hypotenuse is equal to the sum of the squares of the legs.

The Pythagorean Theorem is commonly represented as follows:

a² + b² = c²

Where:

a and b are the lengths of the two legs of the right triangle
c is the length of the hypotenuse of the right triangle

To find the length of a side of a right triangle using the Pythagorean Theorem, you need to have information about the other two sides. Here are the steps to follow:

Step 1: Identify the legs and the hypotenuse of the right triangle

The legs are the two sides that form the right angle.
The hypotenuse is the longest side of the right triangle that is opposite to the right angle.

Step 2: Label the lengths of the legs and the hypotenuse

Use variables such as a, b, and c to represent the lengths of the legs and the hypotenuse.

Step 3: Write the Pythagorean Theorem equation

Substitute the values of a, b, and c into the equation a² + b² = c².
Square the lengths of the legs, add them together, and then take the square root of the result to find the length of the hypotenuse.
Or square the length of the hypotenuse and subtract the square of the known leg to find the length of the unknown leg.

Example of Pythagorean Theorem

Find the length of the hypotenuse c in the right triangle shown below:

Solution:

Step 1: Identify the legs and the hypotenuse

The legs are a and b.
The hypotenuse is c.

Step 2: Label the lengths of the legs and the hypotenuse

Let a = 3 and b = 4.
Let c be the unknown length.

Step 3: Write the Pythagorean Theorem equation

Substitute a, b, and c into the equation a² + b² = c².
3² + 4² = c²
9 + 16 = c²
25 = c²
c = √25
c = 5

Therefore, the length of the hypotenuse c is 5 units.