A piece of wire is bent into the shape of a triangle. Two sides have length 16 inches and 23 inches. The angle between these two sides is 40°.

Question

What is the length of the third side to the nearest hundredth of an inch?

The length of third side is approximately __?__ inches.

Answer 1

third side is approximately 14,87 inches

2 sides are 16 and 23, angle between them is 40 degrees

construct a right triangle with hypotenuse =16

base = 16cos40 = about 12.26

height = 16sin40 = about 10.28

construct another right triangle adjoining the 1st right triangle to the right with same height = 10.28 but with base 23-12.26 = about 10.74

take the square root of sum of squares = [10.74^2 + 10.28^2]^1/2 = 14.87

third side is 14.87 inches the hypotenuse of the 2nd right triangle

or get the same answer using the law of cosines

Answer 2

Hello Zaira!

The basic intuition behind this type of problem is that in a triangle, a side length has a direct relationship to the measure of the opposite angle. The shortest side will be across from the smallest angle, and the largest side will be across from the largest angle.

Depending on what information is missing, you can use the Law of Sines or the Law of Cosines. Based on the missing information in this one, you will use the Law of Cosines.

(If you had the measure of a side and the length of its opposite angle as your givens, you would use the Law of Sines, but that's exactly what you're missing here!)

Here is the equation for the Law of Cosines:

c² = a² + b² - 2ab * cos(C)

If you plug in the information given in this problem, you will get:

c² = (16)² + (23)² - 2(16)(23)*cos(40)

Which solves to...

c² = 221.191289871

...and then...

c = 14.8725011

I hope that helps!

Katherine

Answer 3

This is Law of Cosines. If x = missing side, then

x = √(16²+23²-2(16)(23)cos(40))