Question

question 9 of 10
calculate the side lengths a and b to two decimal places.

a. a = 15.68 and b = 19.58
b. a = 16 and b = 17
c. a = 6.23 and b = 4.45
d. a = 14.71 and b = 17.56

Explanation:

Step1: Find angle at C

Sum of angles in a triangle is \(180^\circ\). So, \(\angle C = 180^\circ - 50^\circ - 107^\circ = 23^\circ\).

Step2: Apply Law of Sines

Law of Sines: \(\frac{a}{\sin A}=\frac{b}{\sin B}=\frac{c}{\sin C}\), where \(c = 8\) (side AB), \(A = 50^\circ\), \(B = 107^\circ\), \(C = 23^\circ\).

For side \(a\) (opposite \(\angle A\)):
\(\frac{a}{\sin 50^\circ}=\frac{8}{\sin 23^\circ}\)
\(a=\frac{8\sin 50^\circ}{\sin 23^\circ}\)
\(\sin 50^\circ\approx0.7660\), \(\sin 23^\circ\approx0.3907\)
\(a\approx\frac{8\times0.7660}{0.3907}\approx\frac{6.128}{0.3907}\approx15.68\)

For side \(b\) (opposite \(\angle B\)):
\(\frac{b}{\sin 107^\circ}=\frac{8}{\sin 23^\circ}\)
\(\sin 107^\circ=\sin(73^\circ)\approx0.9563\)
\(b=\frac{8\sin 107^\circ}{\sin 23^\circ}\approx\frac{8\times0.9563}{0.3907}\approx\frac{7.6504}{0.3907}\approx19.58\)

Answer:

A. \(a = 15.68\) and \(b = 19.58\)