Chapter 8.4, Problem 70PFA

To determine

To find: the Square of the length of the hypotenuse.

Answer to Problem 70PFA

  $32x^2+16x+2$

Explanation of Solution

Given:

The area of an isosceles right triangle is $8x^2+4x+0.5$

Calculation:

Let the hypotenuse of right isosceles right triangle is $c$ and the other two equal sides are $a$

  

Thus, the area of the right isosceles right triangle is given by

  $\begin{array}{l}\frac{1}{2}\cdot a\cdot a\\ =\frac{1}{2}{a}^2\end{array}$

This should be equal to $8x^2+4x+0.5$

  $\begin{array}{l}\frac{1}{2}{a}^2=8x^2+4x+0.5\\ a^2=16x^2+8x+1\end{array}$

Now, using the Pythagoras theorem,

  $\begin{array}{l}{c}^2=a^2+a^2\\ c^2=2a^2\\ c^2=2\left(16x^2+8x+1\right)\\ c^2=32x^2+16x+2\end{array}$

Therefore, the square of the hypotenuse is given by

  $32x^2+16x+2$