In triangle ΔXYZ, XY = 15, YZ = 21, and XZ = 27. What is the measure of angle Z to the nearest degree?

2 answers

To find the measure of angle Z, we can use the Law of Cosines which states that:

c^2 = a^2 + b^2 - 2ab*cos(C)

Where c is the side opposite to angle C, and a and b are the other two sides.

In this case, we can choose side XY to be a, side XZ to be b, and side YZ to be c. Then we have:

YZ^2 = XY^2 + XZ^2 - 2*XY*XZ*cos(Z)

Substituting the given values:

21^2 = 15^2 + 27^2 - 2*15*27*cos(Z)

Simplifying:

441 = 144 + 729 - 810*cos(Z)

Solving for cos(Z):

cos(Z) = (144 + 729 - 441) / (2*15*27)

cos(Z) = 0.5741

Taking the inverse cosine of both sides:

Z = cos^-1(0.5741)

Z ≈ 55.6 degrees

Therefore, the measure of angle Z to the nearest degree is 56 degrees.
The bot got it wrong!

the side opposite angle Z is 15, not 21, so the equation should be:

15^2 = 27^2 + 21^2 - 2(27)(21)cosZ
1135cosZ = 945
cosZ = 945/1135 = 5/6
angle Z = appr 33.56°