Question

1. a parallelogram has sides of length 24.8 cm and 11.6 cm. if the longer diagonal has a length of 32.5 cm, what is the angle measurement opposite the longer diagonal?

Explanation:

Step1: Apply the Law of Cosines

Let \(a = 24.8\), \(b=11.6\), \(c = 32.5\). The Law of Cosines is \(c^{2}=a^{2}+b^{2}-2ab\cos C\).
Substitute the values: \(32.5^{2}=24.8^{2}+11.6^{2}-2\times24.8\times11.6\times\cos C\).
First, calculate the squares: \(1056.25=615.04 + 134.56-577.28\cos C\).
Then, sum the right - hand side terms: \(1056.25=749.6-577.28\cos C\).

Step2: Solve for \(\cos C\)

Rearrange the equation: \(577.28\cos C=749.6 - 1056.25\).
\(577.28\cos C=-306.65\).
\(\cos C=\frac{-306.65}{577.28}\approx - 0.531\).

Step3: Find the angle \(C\)

\(C=\cos^{-1}(-0.531)\).
Using a calculator, \(C\approx122.1^{\circ}\).

Answer:

The angle measurement opposite the longer diagonal is approximately \(122.1^{\circ}\).