Question

what is the measure of the largest angle of the triangle? 15 18?° 26 round to the nearest hundredth.

Explanation:

Step1: Identify the largest side and opposite angle

In a triangle, the largest angle is opposite the longest side. Here, the longest side is 26, so we use the Law of Cosines to find the angle opposite to it. The Law of Cosines formula is \( c^2 = a^2 + b^2 - 2ab\cos(C) \), where \( c \) is the side opposite angle \( C \), and \( a \) and \( b \) are the other two sides. Let \( a = 15 \), \( b = 18 \), \( c = 26 \). We need to solve for \( C \).

Rearranging the formula for \( \cos(C) \):
\( \cos(C) = \frac{a^2 + b^2 - c^2}{2ab} \)

Step2: Substitute the values

Substitute \( a = 15 \), \( b = 18 \), \( c = 26 \) into the formula:
\( \cos(C) = \frac{15^2 + 18^2 - 26^2}{2 \times 15 \times 18} \)
First, calculate the squares:
\( 15^2 = 225 \), \( 18^2 = 324 \), \( 26^2 = 676 \)
Then, substitute these values:
\( \cos(C) = \frac{225 + 324 - 676}{540} \)
Calculate the numerator:
\( 225 + 324 = 549 \), \( 549 - 676 = -127 \)
So, \( \cos(C) = \frac{-127}{540} \approx -0.235185 \)

Step3: Find the angle

To find \( C \), take the arccosine of \( -0.235185 \):
\( C = \arccos(-0.235185) \)
Using a calculator, we find:
\( C \approx 103.62^\circ \) (rounded to the nearest hundredth)

Answer:

\( 103.62^\circ \)