Find the missing parts of the triangle.
-$$\begin{array} { l }
\mathrm { B } = 106.3 ^ { \circ } \
\mathrm { a } = 298 \mathrm {~cm} \
\mathrm {~b} = 1353 \mathrm {~cm}
\end{array}$$
If necessary, round angles the nearest tenth and side lengths to the nearest $\mathrm { cm }$.
A) $\mathrm { A } = 12.2 ^ { \circ } , \mathrm { C } = 61.5 ^ { \circ } , \mathrm { c } = 1239 \mathrm {~cm}$
B) $\mathrm { A } = 61.5 ^ { \circ } , \mathrm { C } = 12.2 ^ { \circ } , \mathrm { c } = 1239 \mathrm {~cm}$
C) no such triangle
D) $\mathrm { A } = 15.2 ^ { \circ } , \mathrm { C } = 58.5 ^ { \circ } , \mathrm { c } = 1239 \mathrm {~cm}$

Question

Find the missing parts of the triangle.
-$$\begin{array} { l } 
\mathrm { B } = 106.3 ^ { \circ } \
\mathrm { a } = 298 \mathrm {~cm} \
\mathrm {~b} = 1353 \mathrm {~cm}
\end{array}$$
If necessary, round angles the nearest tenth and side lengths to the nearest $\mathrm { cm }$.
A) $\mathrm { A } = 12.2 ^ { \circ } , \mathrm { C } = 61.5 ^ { \circ } , \mathrm { c } = 1239 \mathrm {~cm}$
B) $\mathrm { A } = 61.5 ^ { \circ } , \mathrm { C } = 12.2 ^ { \circ } , \mathrm { c } = 1239 \mathrm {~cm}$
C) no such triangle
D) $\mathrm { A } = 15.2 ^ { \circ } , \mathrm { C } = 58.5 ^ { \circ } , \mathrm { c } = 1239 \mathrm {~cm}$

Quizplus · Accepted Answer

The Law of Sines can be used to find the missing parts of the triangle.$$\frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C}$$We are given that $a = 298 \mathrm{~cm}$, $b = 1353 \mathrm{~cm}$, and $B = 106.3^\circ$. We can use the Law of Sines to find $A$:$$\frac{298}{\sin A} = \frac{1353}{\sin 106.3^\circ}$$$$\sin A = \frac{298 \sin 106.3^\circ}{1353} \approx 0.212$$$$A \approx \sin^{-1}(0.212) \approx 12.2^\circ$$We can then use the fact that the sum of the angles in a triangle is $180^\circ$ to find $C$:$$C = 180^\circ - A - B = 180^\circ - 12.2^\circ - 106.3^\circ \approx 61.5^\circ$$Finally, we can use the Law of Sines again to find $c$:$$\frac{1353}{\sin 106.3^\circ} = \frac{c}{\sin 61.5^\circ}$$$$c = \frac{1353 \sin 61.5^\circ}{\sin 106.3^\circ} \approx 1239 \mathrm{~cm}$$Therefore, the missing parts of the triangle are $A \approx 12.2^\circ$, $C \approx 61.5^\circ$, and $c \approx 1239 \mathrm{~cm}$.

Find the Missing Parts of the Triangle B=106.3∘a=298 cm b=1353 cm\begin{array} { l } \mathrm { B } = 106.3 ^ { \circ } \\\mathrm { a } = 298 \mathrm {~cm} \\\mathrm {~b} = 1353 \mathrm {~cm}\end{array}B=106.3∘a=298 cm b=1353 cm​

Find the Missing Parts of the Triangle $\begin{array} { l } \mathrm { B } = 106.3 ^ { \circ } \\\mathrm { a } = 298 \mathrm {~cm} \\\mathrm {~b} = 1353 \mathrm {~cm}\end{array}$