Question

in triangle abc, m∠a = 25°, m∠c = 61°, and b = 38. solve for the other sides and angle. which measurement is not correct?

a. m∠b = 94°
b. c = 33.3
c. a = 18.4

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 \(m\angle A = 25^{\circ}\) and \(m\angle C=61^{\circ}\):
\(m\angle B=180^{\circ}-25^{\circ}-61^{\circ}=94^{\circ}\).

Step2: Use the Law of Sines

The Law of Sines is \(\frac{a}{\sin A}=\frac{b}{\sin B}=\frac{c}{\sin C}\).
We know \(b = 38\), \(m\angle A = 25^{\circ}\), \(m\angle B=94^{\circ}\), \(m\angle C=61^{\circ}\).
For side \(a\): \(\frac{a}{\sin25^{\circ}}=\frac{38}{\sin94^{\circ}}\), so \(a=\frac{38\times\sin25^{\circ}}{\sin94^{\circ}}\).
\(\sin25^{\circ}\approx0.4226\), \(\sin94^{\circ}\approx0.9976\), then \(a=\frac{38\times0.4226}{0.9976}\approx16.1\).
For side \(c\): \(\frac{c}{\sin61^{\circ}}=\frac{38}{\sin94^{\circ}}\), so \(c=\frac{38\times\sin61^{\circ}}{\sin94^{\circ}}\).
\(\sin61^{\circ}\approx0.8746\), then \(c=\frac{38\times0.8746}{0.9976}\approx33.3\).

Answer:

C. \(a = 18.4\)