Chapter 8.5, Problem 18PSB

To determine

To find values of two sides in terms of x .

Answer to Problem 18PSB

CD is equal to $\frac{25x}{4}$ and AE is equal to $\frac{32x}{5}$ .

Explanation of Solution

Given information:

  $\overleftrightarrow{BE}$ is parallel to $\overleftrightarrow{CD}$ .

  $\begin{array}{l}AB=4x,\\ BC=x,\\ AD=8x,\\ BE=5x\end{array}$

Formula used:

The below property is used:

If two pairs of angles in each triangles are congruent by parallel lines, then corresponding angles are congruent.

Proof:

  

It is given that,

  $\overleftrightarrow{BE}$ is parallel to $\overleftrightarrow{CD}$ .

  $\begin{array}{l}AB=4x,\\ BC=x,\\ AD=8x,\\ BE=5x\end{array}$

If two pairs of angles in each triangles are congruent by parallel lines, then corresponding angles are congruent.

Triangles are similar by AA congruence rule.

  $\begin{array}{l}\frac{AB}{BE}=\frac{AC}{CD}\\ \frac{4x}{5x}=\frac{5x}{CD}\\ (4x)(CD)=25x^2\\ \therefore CD=\frac{25x}{4}\end{array}$

  $\begin{array}{l}\frac{AB}{AC}=\frac{AE}{AD}\\ \frac{4x}{5x}=\frac{AE}{8x}\\ (5x)(AE)=32x^2\\ \therefore AE=\frac{32x}{5}\end{array}$