Question

what is the measure of the largest angle of a triangle having sides of length 20.2, 34.2, and 21.3? ?° round to the nearest hundredth.

Explanation:

Step1: Identify the longest side

In a triangle, the largest angle is opposite the longest side. The sides are \(20.2\), \(34.2\), and \(21.3\). The longest side is \(34.2\), so we'll find the angle opposite to it using the Law of Cosines. The Law of Cosines formula is \(c^{2}=a^{2}+b^{2}-2ab\cos(C)\), where \(C\) is the angle opposite side \(c\). Here, \(c = 34.2\), \(a=20.2\), \(b = 21.3\). We need to solve for \(C\).

Rearranging the Law of Cosines to solve for \(\cos(C)\):
\(\cos(C)=\frac{a^{2}+b^{2}-c^{2}}{2ab}\)

Step2: Substitute the values

Substitute \(a = 20.2\), \(b=21.3\), \(c = 34.2\) into the formula:

First, calculate \(a^{2}\), \(b^{2}\), \(c^{2}\):
\(a^{2}=(20.2)^{2}=20.2\times20.2 = 408.04\)
\(b^{2}=(21.3)^{2}=21.3\times21.3=453.69\)
\(c^{2}=(34.2)^{2}=34.2\times34.2 = 1169.64\)

Then, calculate the numerator: \(a^{2}+b^{2}-c^{2}=408.04 + 453.69-1169.64=861.73 - 1169.64=- 307.91\)

Calculate the denominator: \(2ab=2\times20.2\times21.3 = 2\times430.26 = 860.52\)

So, \(\cos(C)=\frac{-307.91}{860.52}\approx - 0.3578\)

Step3: Find the angle

To find \(C\), we take the arccosine of \(-0.3578\):
\(C=\arccos(- 0.3578)\)

Using a calculator (make sure it's in degree mode), we get \(C\approx110.97^{\circ}\) (rounded to the nearest hundredth)

Answer:

\(110.97\)