Chapter 12, Problem 3CT

To determine

Find the value of $\left(x\right)$ in the diagram shown below.

  

Answer to Problem 3CT

The value of $\left(x\right)$ is $6\sqrt{6}$ .

Explanation of Solution

Given information: The given diagram is shown as,

  

Calculation: From triangle ABC,

  $\begin{array}{c}\angle ACB=180-90-45\\ =45°\end{array}$

Therefore $AB=BC$ because triangle ABC is a isosceles right triangle. Now by Pythagoras Theorem in triangle ABC,

  $\begin{array}{c}AC=\sqrt{{\left(AB\right)}^2+{\left(BC\right)}^2}\\ =\sqrt{{\left(12\right)}^2+{\left(12\right)}^2}\\ =12\sqrt{2}\end{array}$

Now in right triangle CDA, by trigonometric ratios,

  $\begin{array}{c}\cos \left(30\right)=\frac{x}{AC}\\ \cos \left(30\right)=\frac{x}{12\sqrt{2}}\\ x=\left(12\sqrt{2}\right)\cos \left(30\right)\\ x=6\sqrt{6}\end{array}$

Hence the value of $\left(x\right)$ is $6\sqrt{6}$ .