How do you solve the triangle given m∠B = 105°, b = 23, a = 14?

1 Answer
Oct 15, 2016

Start by drawing a diagram.

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Since we have one angle with an opposite side and another side known, we are dealing with an ambiguous case. There are either 0, 1 or 20,1or2 solutions to this triangle.

We start by determining the measure of AA using the law of Sines, since we already know the measure of side aa.

sinA/a = sinB/bsinAa=sinBb

sinA/14 = (sin105˚)/23

A = arcsin((14sin105˚)/23)

A = 36˚

There is only one solution to this triangle, because if it was, then the alternative measure of A would be 180˚ - 36˚ = 144˚, which when added to B, would make the sum of the angles in the triangle exceed 180˚.

We can now use the measure of angles A and B to solve for angle C.

A + B + C = 180˚

36˚ + 105˚ + C = 180˚

C = 39˚

The last step is using this information to reapply the law of sines and determine the length of side c.

sinB/b = sinC/c

(sin105˚)/23 = (sin39˚)/c

c = (23sin39˚)/(sin105˚)

c= 15

In summary

The triangle has the following measures.

•A = 36˚
•a = 14" units"
•B = 105˚
•b = 23" units"
•C = 39˚
•c = 15" units"

Hopefully this helps!