In triangle ΔXYZ, XY = 13, YZ = 20, and XZ = 25. What is the measure of angle Z to the nearest degree?
(1 point)
Responses
59°
59°
62°
62°
31°
31°
74°
To find the measure of angle Z, we can use the Law of Cosines.
The formula is as follows: c^2 = a^2 + b^2 - 2abcos(C), where c is the side opposite angle C.
In this case, side XY is opposite angle Z, so we can write:
25^2 = 13^2 + 20^2 - 2(13)(20)cos(Z)
Simplifying this equation gives us:
625 = 169 + 400 - 520cos(Z)
Combining like terms:
456 = -520cos(Z)
Dividing both sides by -520:
cos(Z) = -456/520
Taking the inverse cosine of both sides:
Z = arccos(-456/520)
Using a calculator to find the arccos of -456/520 gives us a value of approximately 31 degrees.
Therefore, the measure of angle Z to the nearest degree is 31°.