Answer
There are two possible triangles.
The angles of one triangle are as follows:
$A = 61.7^{\circ}, B = 74.6^{\circ},$ and $C = 43.7^{\circ}$
The lengths of the sides are as follows:
$a = 78.9~m, b = 86.4~m,$ and $c = 61.9~m$
The angles of another possible triangle are as follows:
$A = 61.7^{\circ}, B = 105.4^{\circ},$ and $C = 12.9^{\circ}$
The lengths of the sides are as follows:
$a = 78.9~m, b = 86.4~m,$ and $c = 20.0~m$
Work Step by Step
$A = 61.7^{\circ}$
$a = 78.9~m$
$b = 86.4~m$
We can use the law of sines to find angle $B$:
$\frac{a}{sin~A} = \frac{b}{sin~B}$
$sin~B = \frac{b~sin~A}{a}$
$B = arcsin(\frac{b~sin~A}{a})$
$B = arcsin(\frac{86.4~sin~61.7^{\circ}}{78.9})$
$B = arcsin(0.96417)$
$B = 74.6^{\circ}$
We can find angle $C$:
$A+B+C = 180^{\circ}$
$C = 180^{\circ}-A-B$
$C = 180^{\circ}-61.7^{\circ}-74.6^{\circ}$
$C = 43.7^{\circ}$
We can use the law of sines to find the length of side $c$:
$\frac{c}{sin~C} = \frac{a}{sin~A}$
$c = \frac{a~sin~C}{sin~A}$
$c = \frac{(78.9~m)~sin~(43.7^{\circ})}{sin~61.7^{\circ}}$
$c = 61.9~m$
Note that another possible value for angle $B$ is $180^{\circ}-74.6^{\circ} = 105.4^{\circ}$
We can find angle $C$:
$A+B+C = 180^{\circ}$
$C = 180^{\circ}-A-B$
$C = 180^{\circ}-61.7^{\circ}-105.4^{\circ}$
$C = 12.9^{\circ}$
We can use the law of sines to find the length of side $c$:
$\frac{c}{sin~C} = \frac{a}{sin~A}$
$c = \frac{a~sin~C}{sin~A}$
$c = \frac{(78.9~m)~sin~(12.9^{\circ})}{sin~61.7^{\circ}}$
$c = 20.0~m$
