Chapter 5.6, Problem 25E

To determine

To find: The area of the parallelogram.

Answer to Problem 25E

  $\text{Area}=234.8u^2$

Explanation of Solution

Given information: Geometry: The adjacent sides of a parallelogram measure 14 centimeters and 20 centimeters and one angle measures ${57}^{\circ }$ .

Calculation:

We have two formulas. It depends on which information we are given as to which one we use.

If we are given two sides and the included angle we use,

  $k=\frac{1}{2}bc\sin A$

If we are given (or can figure out) three angles and a side we use,

  $k=\frac{1}{2}{b}^2\frac{\sin A\times \sin C}{\sin B}$

In this problem we are going to find the area of one of the two congruent triangles (SSS) that the red diagonal cuts the parallelogram into. We have two sides and an included angle so we use the first equation.

Let b = 14, c = 20 and $\angle A={123}^{\circ }$

  $\begin{array}{l}k=\frac{1}{2}bc\sin A\\ k=\frac{1}{2}\times 14\times 20\sin {123}^{\circ }\\ k=0.5\times 280\times 0.8387\\ k=117.4\end{array}$

This is only one triangle, there are two congruent triangles making up the parallelogram so we need to double this.

  $\text{Area}=117.4\times 2=234.8$

Thus,

  $\text{Area}=234.8u^2$