Chapter 11.2, Problem 29PPS

To determine

To Prove: The theorem $x^2+y^2=z^2$ .

Explanation of Solution

Given:

The given diagram is shown in Figure 1

  

Figure 1

Calculation:

The area of the triangle is of the form,

  $A=\frac{1}{2}bh$

The area of the trapezoid is of the form,

  $A=\frac{1}{2}\left(b_1+b_2\right)h$

From, the given figure the area of the trapezoid is the sum of the area of the triangle as,

  $\begin{array}{l}\frac{1}{2}\left(xy\right)+\frac{1}{2}\left(z\right)\left(z\right)+\frac{1}{2}\left(xy\right)=\frac{1}{2}\left(x+y\right)\left(y+x\right)\\ \frac{1}{2}{z}^2+xy=\frac{1}{2}{x}^2+xy+\frac{1}{2}{y}^2\\ \frac{1}{2}{z}^2=\frac{1}{2}{x}^2+\frac{1}{2}{y}^2\\ z^2=x^2+y^2\end{array}$

Hence, Proved.