Question

calculate the side lengths a and b to two decimal places.
a. a = 16.23 and b = 18.83
b. a = 4.81 and b = 5.33
c. a = 16.23 and b = 17.99
d. a = 17.99 and b = 16.23
e. a = 5.33 and b = 4.81

Explanation:

Step1: Find angle \(C\)

Sum of angles in a triangle is \(180^{\circ}\). So \(C = 180^{\circ}-64^{\circ}-85^{\circ}=31^{\circ}\)

Step2: Use the sine rule \(\frac{a}{\sin A}=\frac{b}{\sin B}=\frac{c}{\sin C}\)

Here \(c = 9.3\), \(A = 64^{\circ}\), \(B = 85^{\circ}\), \(C = 31^{\circ}\)
For \(a\): \(\frac{a}{\sin64^{\circ}}=\frac{9.3}{\sin31^{\circ}}\), so \(a=\frac{9.3\times\sin64^{\circ}}{\sin31^{\circ}}\approx\frac{9.3\times0.8988}{0.5150}\approx16.23\)
For \(b\): \(\frac{b}{\sin85^{\circ}}=\frac{9.3}{\sin31^{\circ}}\), so \(b=\frac{9.3\times\sin85^{\circ}}{\sin31^{\circ}}\approx\frac{9.3\times0.9962}{0.5150}\approx18.83\)

Answer:

A. \(a = 16.23\) and \(b = 18.83\)