Pythagoras Theorem Proof

Pythagoras theorem can be proved in many ways. Some of the most common and widely used methods are the algebraic method and the similar triangles method. Let us have a look at both these methods individually in order to understand the proof of this theorem.

Proof of Pythagorean Theorem Formula using the Algebraic Method

The proof of the Pythagoras theorem can be derived using the algebraic method. For example, let us use the values a, b, and c as shown in the following figure and follow the steps given below:

• Step 1: Arrange four congruent right triangles in the given square PQRS, whose side is a + b. The four right triangles have ‘b’ as the base, ‘a’ as the height and, ‘c’ as the hypotenuse.
• Step 2: The 4 triangles form the inner square WXYZ as shown, with ‘c’ as the four sides.
• Step 3: The area of the square WXYZ by arranging the four triangles is c².
• Step 4: The area of the square PQRS with side (a + b) = Area of 4 triangles + Area of the square WXYZ with side ‘c’. This means (a + b)² = [4 × 1/2 × (a × b)] + c².This leads to a² + b² + 2ab = 2ab + c². Therefore, a² + b² = c². Hence proved.

Pythagorean Theorem Formula Proof using Similar Triangles

Two triangles are said to be similar if their corresponding angles are of equal measure and their corresponding sides are in the same ratio. Also, if the angles are of the same measure, then by using the sine law, we can say that the corresponding sides will also be in the same ratio. Hence, corresponding angles in similar triangles lead us to equal ratios of side lengths.