Consider a triangle ABC like the one below. Suppose that a=68,b=54 , and c=47. (The figure is not drawn to scale.) Solve the triangle.

Question

Consider a triangle ABC like the one below. Suppose that a=26,b=30 , and c=50. (The figure is not drawn to scale.) Solve the triangle.

Answer 1

Hi Andrew,

It looks like you have posted two different triangles. Both can be solved in a similar manner, so I'll start with the first one you provided.

If we know the length of the 3 sides of a triangle, we may utilize the Law of Cosines and then Law of Sines afterward to assist us to find the angle measurements. The Law of Cosines looks similar to the Pythagorean Theorem but it accounts for all triangles, meaning it will help us with oblique (non-right) triangles as well!

Law of Cosines state that (note that there are other arrangements depending on which angle you want to identify first. I am starting by looking for Angle C, but you can choose to look for Angle A or B first):

c² = a² + b² - 2ab*cos C

If you rearrange this algebraically and add 2ab*cos C on both sides and subtract c² on both sides, you will find that:

2ab*cos C = a² + b² - c²

Divide both sides by 2ab, you will find another equation of the Law of Cosines based upon finding angle measurement more directly!

cos C = (a² + b² - c²)/2ab

Now, we know that the length of each of the sides in the triangle are:

a = 68

b = 54

c = 47

If we plug this in, we will get:

cos C = (68² + 54² - 47²)/(2(68)(54))

After taking the arccos of the ratio on both sides, you will get Angle C = 43.5 degrees.

We could use the Law of Cosines again, but using the Law of Sines will be simpler from here. The Law of Sines state that:

sin A ₌ sin B ₌ sin C

a b c

We can find Angle A or B first. Let's choose Angle B and compare that to sin C and side c that we already know the ratio to:

sin B ₌ sin 43.5

54 47

After cross multiplying, we get:

47*sin B = 54*sin(43.5) ----> Divide by 47 on both sides

sin B = 0.790 -----> (Without rounding until the end, take the arcsin of 0.790...)

Angle B = 52.2 degrees

Finally, we know that the sum of interior angles in a triangle is 180 degrees! So now we can just take 180 - Angle C - Angle B to get Angle A!

180 - 43.5 - 52.2 = 84.5 degrees

In summary, the final answers are: (do note I rounded the final angle measurements to the nearest tenth, but definitely follow the rounding rule they specify in the problem if it differs than what I rounded to, and make sure not to round anything until the very end to avoid rounding errors in general!):

Angle A = 84.5 degrees

Angle B = 52.2 degrees

Angle C = 43.5 degrees

This aligns with our expectation that the longest side (a) is across the largest angle (84.5 degrees); the second longest side (b) is across the second largest angle (52.2 degrees) and the smallest side (c) is across the smallest angle (43.5 degrees)!

Any questions, let me know.

Thanks!