Question

question 23 of 25 suppose a triangle has two sides of length 32 and 35, and that the angle between these two sides is 120°. what is the length of the third side of the triangle? a. 33.60 b. 58.04 c. 47.43 d. 53

Explanation:

Step1: Apply the Law of Cosines

Let \(a = 32\), \(b = 35\), and \(\theta=120^{\circ}\). The Law of Cosines formula for finding the third - side \(c\) is \(c^{2}=a^{2}+b^{2}-2ab\cos\theta\).

Step2: Substitute the values

\(\cos120^{\circ}=-\frac{1}{2}\), \(a = 32\), \(b = 35\). Then \(c^{2}=32^{2}+35^{2}-2\times32\times35\times(-\frac{1}{2})\).
First, calculate \(32^{2}=32\times32 = 1024\), \(35^{2}=35\times35 = 1225\), and \(2\times32\times35\times(-\frac{1}{2})=- 1120\).
So \(c^{2}=1024 + 1225+1120\).

Step3: Calculate \(c^{2}\)

\(c^{2}=1024+1225 + 1120=3369\).

Step4: Find \(c\)

\(c=\sqrt{3369}\approx58.04\).

Answer:

B. 58.04