Question

a parallelogram has sides of length 14.7 cm and 11.8 cm. the longer diagonal has length 22.1 cm. find the angle opposite the longer diagonal. what is the degree measure of the angle opposite the longer diagonal?
(round to the nearest tenth as needed.)

Explanation:

Step1: Identify the Law to Use

We can use the Law of Cosines here. For a triangle with sides \(a\), \(b\), and \(c\), and the angle \(\theta\) opposite side \(c\), the Law of Cosines is \(c^{2}=a^{2}+b^{2}-2ab\cos\theta\). In a parallelogram, the diagonal divides it into two triangles. The longer diagonal is opposite the larger angle. Let the sides of the parallelogram be \(a = 14.7\) cm, \(b=11.8\) cm and the diagonal \(c = 22.1\) cm. We need to find the angle \(\theta\) opposite the diagonal \(c\).

Step2: Rearrange the Law of Cosines

From \(c^{2}=a^{2}+b^{2}-2ab\cos\theta\), we can rearrange to solve for \(\cos\theta\):

$$ \cos\theta=\frac{a^{2}+b^{2}-c^{2}}{2ab} $$

Step3: Substitute the Values

Substitute \(a = 14.7\), \(b = 11.8\) and \(c=22.1\) into the formula:
First, calculate \(a^{2}=14.7^{2}=216.09\), \(b^{2}=11.8^{2} = 139.24\), \(c^{2}=22.1^{2}=488.41\)
Then, \(a^{2}+b^{2}-c^{2}=216.09 + 139.24-488.41=216.09+139.24 = 355.33-488.41=- 133.08\)
And \(2ab=2\times14.7\times11.8 = 2\times173.46 = 346.92\)
So, \(\cos\theta=\frac{- 133.08}{346.92}\approx - 0.3836\)

Step4: Find the Angle

To find \(\theta\), we take the arccosine of \(-0.3836\):
\(\theta=\arccos(- 0.3836)\)
Using a calculator, \(\theta\approx112.5^{\circ}\) (rounded to the nearest tenth)

Answer:

\(112.5\)