Question

use the image to answer the question. explain a proof of the pythagorean theorem using the image.

Explanation:

Step1: Calculate the area of the large square

The side length of the large square is \(a + b\). Using the formula for the area of a square \(A = s^2\) (where \(s\) is the side - length), the area of the large square is \((a + b)^2=a^{2}+2ab + b^{2}\).

Step2: Calculate the sum of the areas of the smaller components

The large square is composed of four right - triangles with legs \(a\) and \(b\) and hypotenuse \(c\), and a square with side \(c\).
The area of a single right - triangle is \(A_{\triangle}=\frac{1}{2}ab\). The area of four right - triangles is \(4\times\frac{1}{2}ab = 2ab\). The area of the square with side \(c\) is \(A_{c}=c^{2}\). So the sum of the areas of the smaller components is \(2ab + c^{2}\).

Step3: Set the two area expressions equal

Since the area of the large square is equal to the sum of the areas of its components, we have \(a^{2}+2ab + b^{2}=2ab + c^{2}\).
Subtract \(2ab\) from both sides of the equation: \(a^{2}+b^{2}=c^{2}\), which is the Pythagorean Theorem.

Answer:

By equating the area of the large square \((a + b)^2\) to the sum of the areas of the four right - triangles and the square with side \(c\) (\(2ab + c^{2}\)), and simplifying, we get \(a^{2}+b^{2}=c^{2}\), which is the Pythagorean Theorem.