Chapter 12.8, Problem 13P

To determine

Find the length of CD.

Answer to Problem 13P

60.1 inches

Explanation of Solution

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 :

All the angles can be calculated by the properties of vertically opposite angles, linear angles and sum of all the angles of a triangle is $180°$ .

  

Now, consider the lower triangle:

  

Use Law of Sine:

  $\begin{array}{l}\frac{100}{\sin 116°}=\frac{OA}{\sin 40°}=\frac{OB}{\sin 24°}\\ \Rightarrow \frac{100}{\sin 116°}=\frac{OA}{\sin 40°}\text{ and }\frac{100}{\sin 116°}=\frac{OB}{\sin 24°}\\ \Rightarrow OA=\sin 40°\cdot \frac{100}{\sin 116°}\text{ and OB = }\sin 24°\cdot \frac{100}{\sin 116°}\\ \Rightarrow OA\approx 71.516\text{ and }OB\approx 45.25\end{array}$

Again , use law of sine in the middle triangles:

  $\begin{array}{l}\frac{71.516}{\sin 85°}=\frac{OD}{\sin 31°}\text{ and }\frac{45.25}{\sin 71°}=\frac{OC}{\sin 45°}\\ \Rightarrow OD=\sin 31°\cdot \frac{71.516}{\sin 85°}\text{ and OC = }\sin 45°\cdot \frac{45.25}{\sin 71°}\\ \Rightarrow OD\approx 37\text{ and }OC\approx 33.8\end{array}$

Now, Use Law of Cosine to find CD:

  $\begin{array}{l}C{D}^2=O{D}^2+O{C}^2-2(OD)(OC)\cos \angle COD\\ \Rightarrow C{D}^2={37}^2+{33.8}^2-2(37)(33.8)\cos 116°\\ \Rightarrow CD=\sqrt{2511.44-(2501.2)(\cos 116°)}\\ \Rightarrow CD\approx 60.1\text{ in}\text{.}\end{array}$

So, CD = 60.1 inches