Chapter 7.3, Problem 26WE

To determine

To find the values of $x,y,z$

Answer to Problem 26WE

Thevalues of $x,y\text{and}z$ are ${30}^{\circ },20\sqrt{3}\text{and}24$ respectively.

Explanation of Solution

Given information:

The following two similar polygons are provided

  

Since given two polygons are similar, so their corresponding sides are in equal proportions.

Thus, $\frac{20}{12}=\frac{z}{12\sqrt{3}}=\frac{40}{y}$

  $\begin{array}{l}\Rightarrow \frac{5}{3}=\frac{z}{12\sqrt{3}}=\frac{40}{y}\\ \Rightarrow \frac{5}{3}=\frac{z}{12\sqrt{3}},\frac{5}{3}=\frac{40}{y}\\ \Rightarrow z=\frac{5}{3}\times 12\sqrt{3},y=\frac{3}{5}\times 40\\ \therefore z=20\sqrt{3},y=24.\end{array}$

Since the corresponding sides are equal in proportions, so their corresponding angles are equal.

  $\begin{array}{l}\text{Thus,}{x}^{\circ }+{60}^{\circ }+{90}^{\circ }={180}^{\circ }\\ \Rightarrow x+150=180\\ \Rightarrow x=180-150\\ \therefore x={30}^{\circ }\end{array}$

Hence, thevalues of $x,y\text{and}z$ are ${30}^{\circ },20\sqrt{3}\text{and}24$ respectively.