Consider △ABC, where a = 4, c = 8, and m∠C=74∘.

Question

What are the measures of the missing angles and missing side?

Enter your answer by filling in the boxes. Enter all values rounded to the nearest tenth

m<A

m<B

b

Answer 1

With the information we are given, we have a SSA (side, side, angle) triangle. This means that we have the possibility of 2 solutions - an acute solution and an obtuse solution. We always start with the acute solution and use that information to solve for a possible second solution.

Acute Solution

The first step to this problem is to use the Law of Sines to solve for m∠A.

sin 74/8 = sin A/4

sin A = (sin 74×4)/8

A = sin^-1((sin 74×4)/8)

A = 28.7º

Since we know that the interior angles of a triangle add up to 180º, we can use the two angles we know to solve for the final angle.

m∠B = 180º–74º–28.7º

m∠B = 77.3º

All we have left is to find side b, which can also be found with the Law of Sines.

sin 74/8 = sin 77.3/b

b = (sin 77.3×8)/sin 74

b = 8.1

The acute solutions for this triangle are:

m∠A = 28.7º

m∠B = 77.3º

b = 8.1

Obtuse Solution

Now we have to test for the possibility of a second solution. We are going to use the first angle (m∠A = 28.7º) we found in the acute solution to test this. We find the obtuse angle by subtracting the acute angle from the sum of the interior angles of a triangle.

m∠A = 180º–28.7º

m∠A = 151.3º

If the sum of this angle and the angle we are given (m∠C) is less than 180º, we have a second solution and have to keep solving the problem. If the sum is greater than or equal to 180º, we cannot have a second solution and we are done.

74º+151.3º = 225.3º

225.3º > 180°

The sum is greater than 180°, so we do not have a second solution. The only solution for this triangle is the acute solution.

The acute solutions for this triangle are:

m∠A = 28.7º a = 4

m∠B = 77.3º b = 8.1

m∠C = 74° c = 8