In the triangle shown above, m∠B = 105°, a = 12 in, and c = 8 in. What is the approximate length of side b? A.
257.69 in
B.
14.44 in
C.
15.26 in
D.
16.05 in

Step-by-step explanation:

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

2

solutions to this triangle.

We start by determining the measure of

A

using the law of Sines, since we already know the measure of side

a

.

sin

A

a

=

sin

B

b

sin

A

14

=

sin

105

˚

23

A

=

arcsin

(

14

sin

105

˚

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

.

sin

B

b

=

sin

C

c

sin

105

˚

23

=

sin

39

˚

c

c

=

23

sin

39

˚

sin

105

˚

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!

3 is greater then both of those numbers