In the figure below, $a = 9 , b = 11$, and $c = 12$. Use this information to solve the parallelogram for $\theta$. The diagonals of the parallelogram are represented by $c$ and $d$. Round answer to two decimal places.

A) $107.03 ^ { \circ }$
B) $98.52$
C) $90.00 ^ { \circ }$
D) $92.13$
E) $72.97 ^ { \circ }$

Question

In the figure below, $a = 9 , b = 11$, and $c = 12$. Use this information to solve the parallelogram for $\theta$. The diagonals of the parallelogram are represented by $c$ and $d$. Round answer to two decimal places.

A) $107.03 ^ { \circ }$ 
B) $98.52$ 
C) $90.00 ^ { \circ }$ 
D) $92.13$ 
E) $72.97 ^ { \circ }$

Quizplus · Accepted Answer

In a parallelogram, opposite angles are congruent. Let's call the angle opposite to $b$ as $\alpha$. Then, we know that $\theta + \alpha = 180 ^ {\circ}$. Using the Law of Cosines on triangle $ABE$, we have:
$$c^2 = a^2 + b^2 - 2ab\cos\theta$$
Substituting the given values, we get:
$$12^2 = 9^2 + 11^2 - 2 \cdot 9 \cdot 11 \cdot \cos\theta $$
Solving for $\cos\theta$, we get:
$$\cos\theta = \frac{9^2 + 11^2 - 12^2}{2 \cdot 9 \cdot 11} = -\frac{23}{99}$$
Since $0 ^ {\circ} \leq \theta \leq 180 ^ {\circ}$, and $\cos\theta < 0$, we have:
$$\theta \approx \cos^{-1}\left(-\frac{23}{99}\right) \approx 107.03 ^ {\circ}$$
Therefore, the answer is A.

In the Figure Below a=9,b=11a = 9 , b = 11a=9,b=11

In the Figure Below $a = 9 , b = 11$