Chapter 8.2, Problem 12WE

To determine

To find: x

Answer to Problem 12WE

  $x=6$

Explanation of Solution

Given:

The diagram:

Concept Used:

Theorem 4-1: opposite angles of equal sides in triangle are equal.

Theorem 3-11: The addition of all angles of a triangle is 180.

Theorem 8-2: In right triangle, square of hypogenous is equal to the square of base and perpendicular.

Consider the figure shown below:

Now, consider the triangle $△ABC$ , here $\overline{AB}\cong \overline{BC}$ , so by theorem 4-1, the angles $\angle BAC$ and $\angle ACB$ are congruent.

That is,

  $m\angle ACB=m\angle BAC=45$

Apply theorem 3-11 on triangle $△ABC$ ,

  $\begin{array}{c}m\angle BAC+m\angle ACB+m\angle CBA=180\\ 45+45+m\angle CBA=180\\ 90+m\angle CBA=180\end{array}$

  $\begin{array}{c}m\angle CBA=180-90\\ m\angle CBA=90\end{array}$

Consider the right triangle $△ABC$ .

Now, $\overline{AC}$ is the hypotenuse and has length $6\sqrt{2}$and the lengths of the legs are $BA=x$ and $BC=x$ .

So, by theorem 8-2,

  $\begin{array}{c}A{C}^2=B{A}^2+B{C}^2\\ {\left(6\sqrt{2}\right)}^2=x^2+x^2\\ 72=2x^2\end{array}$

  $\begin{array}{c}{x}^2=\frac{72}{2}\\ x^2=36\end{array}$

Now, take square root of both sides,

  $\begin{array}{c}\sqrt{x^2}=\sqrt{36}\\ x=6\end{array}$