Chapter 14.4, Problem 3MRE

To determine

To find: Values of $x,y,z\&w$

Answer to Problem 3MRE

The values are $x=\frac{9}{2},y=\frac{10}{3},w=\frac{9}{2},z=5$ .

Explanation of Solution

Given information :

  $\overline{ED}\left|\right|\overline{BC}\left|\right|\overline{FG}$

Concept used:

If two objects have the same shape, they are called "similar." When two figures are similar, the ratios of the lengths of their corresponding sides are equal. To determine if the triangles shown are similar, compare their corresponding sides.

As per the given problem

Following diagram is provided.

  

Given: $\overline{ED}\left|\right|\overline{BC}\left|\right|\overline{FG}$

  $\Delta OBC$ similar to $\Delta ODE$ .

  $\begin{array}{l}\frac{OB}{OD}=\frac{BC}{DE}=\frac{CO}{EO}\\ \Rightarrow \frac{3}{x}=\frac{2}{3}=\frac{y}{5}\end{array}$

  $\begin{array}{l}\frac{3}{x}=\frac{2}{3}\\ \Rightarrow x=\frac{9}{2}\\ \frac{2}{3}=\frac{y}{5}\\ y=\frac{10}{3}\end{array}$

  $\Delta OBC$ similar to $\Delta OFG$ .

  $\begin{array}{l}\frac{OB}{OF}=\frac{BC}{FG}=\frac{CO}{GO}\\ \Rightarrow \frac{3}{3+w}=\frac{2}{z}=\frac{y}{y+5}\end{array}$

  $\Rightarrow \frac{\frac{10}{3}}{\frac{10}{3}+5}=\frac{10}{10+15}=\frac{10}{25}=\frac{2}{5}$

  $\begin{array}{l}\frac{3}{3+w}=\frac{2}{5}\\ \Rightarrow 15=6+2w\\ \Rightarrow w=\frac{9}{2}\end{array}$

  $\begin{array}{l}\frac{2}{z}=\frac{2}{5}\\ \Rightarrow z=5\end{array}$

Hence,

The values are $x=\frac{9}{2},y=\frac{10}{3},w=\frac{9}{2},z=5$ .