Chapter 15, Problem 8CR

To determine

To Prove:The product of the intercepted diagonals in trapezoid is equal i.e. $AE•BE=DE•EC$

Explanation of Solution

Given:

  

  $\begin{array}{l}ABCD\text{ is a Trapezoid}\\ AD\left|\right|BC\end{array}$

Formula Used:

  $\begin{array}{l}AAA\text{ Similarity}\\ \left|\right|\text{lines}\text{⇒}\text{Congruent Alternate Interior Angles}\\ \text{Verticle Opposite angles are equal}\end{array}$

Calculation:

  $\begin{array}{l}ABCD\text{ is a Trapezoid}\\ AD\left|\right|BC\\ \begin{array}{l}\angle CAD\cong \angle ACB\\ \angle BDA\cong \angle DBC\end{array}\}\text{⇒}\left|\right|\text{ lines}\text{⇒}\text{Congruent interior angles}\\ \angle AED\cong \angle BEC\text{Vertically oppposite angles}\\ \Delta AED~\Delta CEBAAA\text{ Similarity}\\ \text{⇒}\frac{AE}{EC}=\frac{DE}{BE}\\ \text{⇒}AE•BE=DE•EC\end{array}$

Conclusion:

Hence, proved.