Find the missing parts of the triangle.
-$$\begin{array} { l }
\mathrm { A } = 21 ^ { \circ } \
\mathrm { a } = 35 \mathrm {~km} \
\mathrm {~b} = 50 \mathrm {~km}
\end{array}$$
If necessary, round angles to the nearest whole number and side lengths to the nearest $\mathrm { km }$.
A) $\mathrm { B } _ { 1 } = 128 ^ { \circ } , \mathrm { C } _ { 1 } = 31 ^ { \circ } , \mathrm { c } _ { 1 } = 77 \mathrm {~km}$
B) $\mathrm { B } = 31 ^ { \circ } , \mathrm { C } = 128 ^ { \circ } , \mathrm { c } = 77 \mathrm {~km}$
$$\mathrm { B } _ { 2 } = 10 ^ { \circ } , \mathrm { C } _ { 2 } = 149 ^ { \circ } , \mathrm { c } _ { 2 } = 17 \mathrm {~km}$$
C) no such triangle
D) $\mathrm { B } _ { 1 } = 31 ^ { \circ } , \mathrm { C } _ { 1 } = 128 ^ { \circ } , \mathrm { c } _ { 1 } = 77 \mathrm {~km}$
$$\mathrm { B } _ { 2 } = 149 ^ { \circ } , \mathrm { C } _ { 2 } = 10 ^ { \circ } , \mathrm { c } _ { 2 } = 17 \mathrm {~km}$$

Question

Find the missing parts of the triangle.
-$$\begin{array} { l } 
\mathrm { A } = 21 ^ { \circ } \
\mathrm { a } = 35 \mathrm {~km} \
\mathrm {~b} = 50 \mathrm {~km}
\end{array}$$
If necessary, round angles to the nearest whole number and side lengths to the nearest $\mathrm { km }$.
A) $\mathrm { B } _ { 1 } = 128 ^ { \circ } , \mathrm { C } _ { 1 } = 31 ^ { \circ } , \mathrm { c } _ { 1 } = 77 \mathrm {~km}$
B) $\mathrm { B } = 31 ^ { \circ } , \mathrm { C } = 128 ^ { \circ } , \mathrm { c } = 77 \mathrm {~km}$
$$\mathrm { B } _ { 2 } = 10 ^ { \circ } , \mathrm { C } _ { 2 } = 149 ^ { \circ } , \mathrm { c } _ { 2 } = 17 \mathrm {~km}$$
C) no such triangle
D) $\mathrm { B } _ { 1 } = 31 ^ { \circ } , \mathrm { C } _ { 1 } = 128 ^ { \circ } , \mathrm { c } _ { 1 } = 77 \mathrm {~km}$
$$\mathrm { B } _ { 2 } = 149 ^ { \circ } , \mathrm { C } _ { 2 } = 10 ^ { \circ } , \mathrm { c } _ { 2 } = 17 \mathrm {~km}$$

Quizplus · Accepted Answer

The Law of Sines can be used to find the missing angles and sides. Given $A = 21^\circ$, $a = 35$, and $b = 50$, we can find angle B using the ratio $\frac{\sin A}{a} = \frac{\sin B}{b}$. However, since the sine function can have two different angles with the same sine value (one acute and one obtuse) within the range of 0 to 180 degrees, this leads to two possible triangles, hence the two sets of solutions for angles B and C, and side c. The sum of angles in a triangle is always 180 degrees, which helps in determining the specific angles and the remaining side using the Law of Sines again.

Find the Missing Parts of the Triangle A=21∘a=35 km b=50 km\begin{array} { l } \mathrm { A } = 21 ^ { \circ } \\\mathrm { a } = 35 \mathrm {~km} \\\mathrm {~b} = 50 \mathrm {~km}\end{array}A=21∘a=35 km b=50 km​

Find the Missing Parts of the Triangle $\begin{array} { l } \mathrm { A } = 21 ^ { \circ } \\\mathrm { a } = 35 \mathrm {~km} \\\mathrm {~b} = 50 \mathrm {~km}\end{array}$