Chapter 12.8, Problem 15WE

To determine

Find the length of $\overline{CD}$ .

Answer to Problem 15WE

12.03

Explanation of Solution

Given:

In quadrilateral ABCD, $\angle 1=30°,\angle 2=80°,\angle B=80°$ , AD = 8 , and AB = 10.

Formula Used:

Law of Cosine :

  $a^2=b^2+c^2-2bc\cos A$

Law of Sine:

  $\frac{a}{\sin A}=\frac{b}{\sin B}=\frac{c}{\sin C}$

  

Calculation:

Represent the situation and solve :

  

  $\angle ACB=180°-30°-80°=70°$

Use Law of Sine in triangle ABC:

  $\begin{array}{l}\frac{AC}{\sin 80°}=\frac{10}{\sin 70°}\\ \Rightarrow AC=\sin 80°\cdot \frac{10}{\sin 70°}\mathrm{.................}\text{[multiply each side by sin80}°]\\ \Rightarrow AC\approx 10.48\end{array}$

Now, Use Law of Cosine to find CD:

  $\begin{array}{l}C{D}^2=A{C}^2+A{D}^2-2(AC)(AD)\cos \angle CAD\\ \Rightarrow C{D}^2={10.48}^2+8^2-2(10.48)(8)\cos 80°\\ \Rightarrow CD\approx 12.03\end{array}$

So, the length of $\overline{CD}$ is about 12.03 .