Question

solve △abc if a = 5cm, b = 13cm, and c = 9cm. 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: Use the Law of Cosines to find angle \(A\)

The Law of Cosines formula is \(\cos A=\frac{b^{2}+c^{2}-a^{2}}{2bc}\).
Substitute \(a = 5\), \(b = 13\), \(c = 9\) into the formula:
\(\cos A=\frac{13^{2}+9^{2}-5^{2}}{2\times13\times9}=\frac{169 + 81-25}{234}=\frac{225}{234}\approx0.9615\)
Then \(A=\cos^{-1}(0.9615)\approx15.9^{\circ}\)

Step2: Use the Law of Cosines to find angle \(C\)

The Law of Cosines formula is \(\cos C=\frac{a^{2}+b^{2}-c^{2}}{2ab}\)
Substitute \(a = 5\), \(b = 13\), \(c = 9\) into the formula:
\(\cos C=\frac{5^{2}+13^{2}-9^{2}}{2\times5\times13}=\frac{25+169 - 81}{130}=\frac{113}{130}\approx0.8692\)
Then \(C=\cos^{-1}(0.8692)\approx29.9^{\circ}\)

Step3: Use the angle - sum property of a triangle (\(A + B+C=180^{\circ}\)) to find angle \(B\)

\(B=180^{\circ}-(A + C)\)
Substitute \(A\approx15.9^{\circ}\) and \(C\approx29.9^{\circ}\)
\(B=180-(15.9 + 29.9)=134.2^{\circ}\)

Answer:

\(A\approx15.9^{\circ}\), \(B\approx134.2^{\circ}\), \(C\approx29.9^{\circ}\)