Chapter 12.8, Problem 16WE

To determine

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

Answer to Problem 16WE

8.78

Explanation of Solution

Given:

In quadrilateral ABCD, $\angle B=110°,\angle D=75°,\angle 3=35°$ , AB = 5 , and BC = 6.

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 :

  

  $\angle CAD=180°-35°-75°=70°$

Use Law of Cosine to find ABC:

  $\begin{array}{l}A{C}^2=A{B}^2+B{C}^2-2(AB)(BC)\cos \angle ABC\\ \Rightarrow A{C}^2=5^2+6^2-2(5)(6)\cos 110°\\ \Rightarrow AC\approx 9.03\end{array}$

Now, use Law of Sine in triangle ACD:

  $\begin{array}{l}\frac{CD}{\sin 70°}=\frac{9.03}{\sin 75°}\\ \Rightarrow CD=\sin 70°\cdot \frac{9.03}{\sin 75°}\mathrm{.................}\text{[multiply each side by sin70}°]\\ \Rightarrow CD\approx 8.78\end{array}$

So, the length of $\overline{CD}$ is about 8.78