Question

what is the perimeter p of \triangle abc to the nearest whole number?
p =

Explanation:

Step1: Find angle \(A\)

The sum of angles in a triangle is \(180^{\circ}\). So, \(A = 180^{\circ}-40^{\circ}-49^{\circ}=91^{\circ}\)

Step2: Use the Law of Sines

The Law of Sines states \(\frac{a}{\sin A}=\frac{b}{\sin B}=\frac{c}{\sin C}\). Let \(a = BC\), \(b = AC = 13\), \(c = AB\)
\(\frac{BC}{\sin A}=\frac{AC}{\sin B}\), so \(BC=\frac{13\times\sin91^{\circ}}{\sin49^{\circ}}\approx\frac{13\times1}{0.7547}\approx17.23\)
\(\frac{AB}{\sin C}=\frac{AC}{\sin B}\), so \(AB=\frac{13\times\sin40^{\circ}}{\sin49^{\circ}}\approx\frac{13\times0.6428}{0.7547}\approx11.07\)

Step3: Calculate the perimeter

\(P=13 + 17.23+11.07 = 41.3\approx41\)

Answer:

\(41\)