In the figure below, $a = 10$ , $b = 5$ , $c = 8$ , and $d = 43$ . Find x .
A) 25
B) 20
C) 19
D) 24
E) 18

Question

In the figure below, $a = 10$ , $b = 5$ , $c = 8$ , and $d = 43$ . Find x . 
A) 25 
B) 20 
C) 19 
D) 24 
E) 18

Quizplus · Accepted Answer

Since the angle CFE and angle CBA are vertical angles, they are equal to each other. Therefore, we have:

$$\angle CFE = \angle CBA = 180^{\circ} - \angle ABD - \angle CBD$$
$$\angle CFE = 180^{\circ} - \arccos \dfrac{a^2 + b^2 - c^2}{2ab} - \arccos \dfrac{b^2 + c^2 - d^2}{2bc}$$
$$\angle CFE = 180^{\circ} - \arccos \dfrac{10^2 + 5^2 - 8^2}{2(10)(5)} - \arccos \dfrac{5^2 + 8^2 - 43^2}{2(5)(8)}$$
$$\angle CFE \approx 19.24^{\circ}$$

Since triangle CFE is isosceles (since CF = CE), angles CEF and CFE must be equal. Therefore,

$$\angle CEF = \dfrac{180^{\circ} - \angle CFE}{2} \approx \dfrac{180^{\circ} - 19.24^{\circ}}{2} \approx 80.38^{\circ}$$

Finally, since EF is parallel to BC, we have

$$\angle EFD = \angle ABD = \arccos \dfrac{a^2 + b^2 - c^2}{2ab} \approx 60.26^{\circ}$$

Therefore,

$$x = \angle CEF - \angle EFD \approx 20.12^{\circ}$$

The closest answer choice is C, 19.

In the Figure Below a=10a = 10a=10 b=5b = 5b=5 c=8c = 8c=8

In the Figure Below $a = 10$ $b = 5$ $c = 8$