Question

solve the following triangle.
a = 70°, b = 70°, c = 10
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 \( C=180^{\circ}-A - B \).
Substituting \( A = 70^{\circ}\) and \( B = 70^{\circ}\), we get \( C=180^{\circ}-70^{\circ}-70^{\circ}=40^{\circ}\).

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

The Law of Sines states that \(\frac{a}{\sin A}=\frac{c}{\sin C}\).
We know \( A = 70^{\circ}\), \( C = 40^{\circ}\), and \( c = 10\).
So \( a=\frac{c\sin A}{\sin C}\).
Substituting the values: \( a=\frac{10\times\sin70^{\circ}}{\sin40^{\circ}}\).
Since \(\sin70^{\circ}\approx0.9397\) and \(\sin40^{\circ}\approx0.6428\), \( a=\frac{10\times0.9397}{0.6428}\approx14.62\).

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

Since \( A = B = 70^{\circ}\), by the Law of Sines \(\frac{a}{\sin A}=\frac{b}{\sin B}\).
Because \(\sin A=\sin B\) (since \(A = B\)), then \( a = b\). So \( b\approx14.62\).

Answer:

\( C = 40^{\circ}\), \( a\approx14.62\), \( b\approx14.62\)