Question

solve △abc if m∠a = 100°, m∠c = 23°, and c = 250 in. round each side and angle to the nearest tenth as needed. you must show all work for finding the missing sides and angles of the triangle. you must show all steps and your exact, calculator ready value before providing your rounded answer.

Explanation:

Step1: Find \(m\angle B\)

The sum of angles in a triangle is \(180^{\circ}\). So, \(m\angle B=180^{\circ}-m\angle A - m\angle C\).
Substitute \(m\angle A = 100^{\circ}\) and \(m\angle C=23^{\circ}\):
\(m\angle B=180^{\circ}-100^{\circ}-23^{\circ}=57^{\circ}\)

Step2: Use the Law of Sines \(\frac{a}{\sin A}=\frac{b}{\sin B}=\frac{c}{\sin C}\)

First, find \(a\) using \(\frac{a}{\sin A}=\frac{c}{\sin C}\)
\(a=\frac{c\sin A}{\sin C}\)
Substitute \(c = 250\), \(\sin A=\sin(100^{\circ})\approx0.985\), \(\sin C=\sin(23^{\circ})\approx0.391\)
\(a=\frac{250\times0.985}{0.391}\approx630.4\)

Step3: Find \(b\) using \(\frac{b}{\sin B}=\frac{c}{\sin C}\)

\(b=\frac{c\sin B}{\sin C}\)
Substitute \(c = 250\), \(\sin B=\sin(57^{\circ})\approx0.839\), \(\sin C=\sin(23^{\circ})\approx0.391\)
\(b=\frac{250\times0.839}{0.391}\approx536.3\)

Answer:

\(m\angle B = 57^{\circ}\), \(a\approx630.4\) in, \(b\approx536.3\) in