Question

find x in the following triangle:

Explanation:

Step1: Find the length of the base segment adjacent to \(63^{\circ}\)

Using the cosine function: \(\cos(63^{\circ})=\frac{\text{adjacent}}{\text{hypotenuse}}\).
Let the adjacent side be \(a_1\), hypotenuse \( = 27\).
\(a_1 = 27\times\cos(63^{\circ})\)
\(a_1\approx27\times0.4540 = 12.258\)

Step2: Find the length of the base segment adjacent to \(39^{\circ}\)

Using the cosine function: \(\cos(39^{\circ})=\frac{\text{adjacent}}{\text{hypotenuse}}\).
Let the adjacent side be \(a_2\), hypotenuse \( = 51\).
\(a_2 = 51\times\cos(39^{\circ})\)
\(a_2\approx51\times0.7771=39.6321\)

Step3: Calculate \(x\)

\(x=a_1 + a_2\)
\(x\approx12.258+39.6321 = 51.8901\approx51.9\)

Answer:

\(x\approx51.9\)