Chapter 8.4, Problem 16PSB

To determine

Prove that $AE\cdot CD=EC\cdot AB$ .

Answer to Problem 16PSB

Proved $AE\cdot CD=EC\cdot AB$ .

Explanation of Solution

Given information:

  

  $ABCD$ trapezoid with bases $\overline{AB}and\overline{CD}$ .

Calculation:

It is given that,

  $ABCD$ Trapezoid with bases $\overline{AB}and\overline{CD}$ .

Trapezoid have base segment are parallel to each other,

  $DC\left|\right|AB$

If two lines are parallel and cut by transversal then corresponding alternate interior angles are congruent.

  $\begin{array}{l}DC\left|\right|AB\\ \angle DCE\cong \angle BAE\\ \angle DEC\cong \angle AEB\left[verticalcongrientangles\right]\\ \Delta CED=\Delta AEB\left[AA\right]\\ AE/EC=AB/CD\left[correspondingsidelengthareproportional\right]\\ AE\cdot CD=EC\cdot AB\left[extreamproducttheorem\right]\end{array}$

Hence, proved $AE\cdot CD=EC\cdot AB$