Find the missing parts of the triangle. Round to the nearest tenth when necessary or to the nearest minute as appropriate.
-$$\begin{array} { l }
\mathrm { C } = 105.8 ^ { \circ } \
\mathrm { a } = 4.7 \mathrm {~km} \
\mathrm {~b} = 8.3 \mathrm {~km}
\end{array}$$
A) No triangle satisfies the given conditions.
B) $\mathrm { c } = 16.4 \mathrm {~km} , \mathrm {~A} = 23.2 ^ { \circ } , \mathrm { B } = 51 ^ { \circ }$
C) $\mathrm { c } = 10.6 \mathrm {~km} , \mathrm {~A} = 25.2 ^ { \circ } , \mathrm { B } = 49 ^ { \circ }$
D) $\mathrm { c } = 13.5 \mathrm {~km} , \mathrm {~A} = 27.2 ^ { \circ } , \mathrm { B } = 47 ^ { \circ }$

Question

Find the missing parts of the triangle. Round to the nearest tenth when necessary or to the nearest minute as appropriate.
-$$\begin{array} { l } 
\mathrm { C } = 105.8 ^ { \circ } \
\mathrm { a } = 4.7 \mathrm {~km} \
\mathrm {~b} = 8.3 \mathrm {~km}
\end{array}$$
A) No triangle satisfies the given conditions.
B) $\mathrm { c } = 16.4 \mathrm {~km} , \mathrm {~A} = 23.2 ^ { \circ } , \mathrm { B } = 51 ^ { \circ }$
C) $\mathrm { c } = 10.6 \mathrm {~km} , \mathrm {~A} = 25.2 ^ { \circ } , \mathrm { B } = 49 ^ { \circ }$
D) $\mathrm { c } = 13.5 \mathrm {~km} , \mathrm {~A} = 27.2 ^ { \circ } , \mathrm { B } = 47 ^ { \circ }$

Quizplus · Accepted Answer

To solve this, we can use the Law of Sines: $\frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C}$. Given $C = 105.8^\circ$, $a = 4.7$ km, and $b = 8.3$ km, we can find angle $A$ or $B$ first using the relationship between the sides and angles, and then use the Law of Cosines or continue with the Law of Sines to find side $c$. The sum of angles in a triangle is $180^\circ$, so after finding one of the other angles, the third angle can be calculated. The correct values should satisfy these principles, and among the options given, only the set of values in option C can be derived correctly using these laws, considering the rounding instructions.

Find the Missing Parts of the Triangle C=105.8∘a=4.7 km b=8.3 km\begin{array} { l } \mathrm { C } = 105.8 ^ { \circ } \\\mathrm { a } = 4.7 \mathrm {~km} \\\mathrm {~b} = 8.3 \mathrm {~km}\end{array}C=105.8∘a=4.7 km b=8.3 km​

Find the Missing Parts of the Triangle $\begin{array} { l } \mathrm { C } = 105.8 ^ { \circ } \\\mathrm { a } = 4.7 \mathrm {~km} \\\mathrm {~b} = 8.3 \mathrm {~km}\end{array}$