Chapter 11.6, Problem 10P

To determine

To prove: that the triangles with the sides of given lengths is the sides of the right angle triangle.

Answer to Problem 10P

The equation is proved.

Explanation of Solution

Given:

The sides of the triangle are $x^2+y^2,2xy,\text{ and }{x}^2-y^2$ .

Calculation:

According to the question, the sides of the triangle are $x^2+y^2,2xy,\text{ and }{x}^2-y^2$ .

  

Here, from right triangle ABC, AC is the hypotenuse. Therefore,

By using Pythagoras theorem, $A{C}^2=A{B}^2+B{C}^2$ Thus,

  $\begin{array}{l}A{C}^2=A{B}^2+B{C}^2\\ {\left(x^2+y^2\right)}^2={\left(2xy\right)}^2\text{+}{\left(x^2-y^2\right)}^2\left\{\text{substitutes the given value}\right\}\\ x^4+2x^2{y}^2+y^4=4x^2{y}^2+x^4-2x^2{y}^2+y^4\\ 2x^2{y}^2=4x^2{y}^2-2x^2{y}^2\{\text{remove }{x}^4\text{ and }{y}^4\text{ from the sides}}\\ 2x^2{y}^2=2x^2{y}^2\{\text{true}}\end{array}$

Hence, the equation is proved.