Chapter 11, Problem 33RP

To determine

To calculate: The area of trapezoid ABEF if side AF is 2.

Answer to Problem 33RP

The area of trapezoid ABEF is $247.68$ .

Explanation of Solution

Given information:

Side AF = 2,

P(ABEF )=70,

Angle B $={60}^{\circ }$

Formula used:

Area of trapezoid: $A=\frac{1}{2}\times (b_1+b_2)\times h$

  $b_1$ and $b_2$ are bases of trapezoid.

h = height of trapezoid.

Calculation:

  

If AF= 2, then CD = 2.

Triangles ABC and FED are ${30}^{\circ }{60}^{\circ }{90}^{\circ }\Delta \text{'}s$

Two triangles are congruent by HL congruence rule.

  $\Delta ABC\cong \Delta FED$

If BC = DE = x , then

AB = FE = 2x

Perimeter is sum of all sides.

  $\begin{array}{l}P=70\\ AF+FE+DE+CD+BC+AB=70\\ 2+2x+x+2+x+2x=70\\ 4+6x=70\\ 6x=66\\ x=11\end{array}$

In ${30}^{\circ }{60}^{\circ }{90}^{\circ }$ triangle ABC ,

  $\begin{array}{l}\tan {30}^{\circ }=\frac{oppositeside}{adjacentside}\\ \frac{1}{\sqrt{3}}=\frac{x}{AC}\\ AC=x\sqrt{3}\end{array}$

Height $AC=11\sqrt{3}$

Base 1 = 2

Base 2 = BC + CD + DE

Base 2 $=x+2+x=24$

Area of Trapezoid:

  $\begin{array}{l}A=\frac{1}{2}\times (b_1+b_2)\times h\\ A=\frac{1}{2}\times (2+24)\times 11\sqrt{3}\\ A=13\times 11\sqrt{3}\\ A=143\sqrt{3}\\ A=247.68\end{array}$