Question

solve the following triangle.
a = 50°, b = 50°, c = 9
c ≈ □°
(simplify your answer.)
a ≈ □
(type an integer or decimal rounded to two decimal places as needed.)
b ≈ □
(type an integer or decimal rounded to two decimal places as needed.)

Explanation:

Step1: Find angle \(C\)

The sum of angles in a triangle is \(180^{\circ}\). So \(C = 180^{\circ}-A - B\).
Substitute \(A = 50^{\circ}\) and \(B = 50^{\circ}\):
\(C=180^{\circ}-50^{\circ}-50^{\circ}=80^{\circ}\)

Step2: Use the Law of Sines to find \(a\)

The Law of Sines is \(\frac{a}{\sin A}=\frac{c}{\sin C}\).
We know \(A = 50^{\circ}\), \(C = 80^{\circ}\), and \(c = 9\).
So \(a=\frac{c\sin A}{\sin C}\)
\(\sin50^{\circ}\approx0.7660\), \(\sin80^{\circ}\approx0.9848\)
\(a=\frac{9\times0.7660}{0.9848}=\frac{6.894}{0.9848}\approx6.91\)

Step3: Use the Law of Sines to find \(b\)

Since \(A = B = 50^{\circ}\), by the Law of Sines \(\frac{a}{\sin A}=\frac{b}{\sin B}\), and because \(\sin A=\sin B\), we have \(a = b\)
So \(b\approx6.91\)

Answer:

\(C = 80^{\circ}\), \(a\approx6.91\), \(b\approx6.91\)