To solve this problem, we will use trigonometric relations involving the Law of Cosines and the Law of Sines. Follow these steps:
1. Calculate YZ using the Law of Cosines:
In triangle XYZ, we know:
- m∠ZY = 75°
- XY = 10
- XZ = 13
The Law of Cosines states:
[tex]\[ c^2 = a^2 + b^2 - 2ab \cdot \cos(C) \][/tex]
Here, [tex]\( c = YZ \)[/tex], [tex]\( a = XY = 10 \)[/tex], [tex]\( b = XZ = 13 \)[/tex], and [tex]\( C = m∠ZY = 75° \)[/tex].
So,
[tex]\[ YZ^2 = XY^2 + XZ^2 - 2 \cdot XY \cdot XZ \cdot \cos(m∠ZY) \][/tex]
[tex]\[ YZ^2 = 10^2 + 13^2 - 2 \cdot 10 \cdot 13 \cdot \cos(75°) \][/tex]
First, calculate the cosine of 75°:
[tex]\[ \cos(75°) \approx 0.2588 \][/tex]
Now plug in these values:
[tex]\[ YZ^2 = 100 + 169 - 2 \cdot 10 \cdot 13 \cdot 0.2588 \][/tex]
[tex]\[ YZ^2 = 269 - 67.328 \][/tex]
[tex]\[ YZ^2 = 201.672 \][/tex]
Taking the square root to get YZ:
[tex]\[ YZ = \sqrt{201.672} \approx 14.2 \][/tex]
2. Calculate the measure of angle ZZ using the Law of Sines:
Next, we use the Law of Sines, which states:
[tex]\[ \frac{\sin(A)}{a} = \frac{\sin(B)}{b} = \frac{\sin(C)}{c} \][/tex]
Let's denote angle X as ∠ZX (since this is the corner we're actually solving for).
[tex]\[ \frac{\sin(X)}{XZ} = \frac{\sin(75°)}{YZ} \][/tex]
Plug in known values:
[tex]\[ \frac{\sin(X)}{13} = \frac{\sin(75°)}{14.2} \][/tex]
Calculate sine of 75°:
[tex]\[ \sin(75°) \approx 0.9659 \][/tex]
Now,
[tex]\[ \frac{\sin(X)}{13} = \frac{0.9659}{14.2} \][/tex]
[tex]\[ \sin(X) = 13 \cdot \frac{0.9659}{14.2} \][/tex]
[tex]\[ \sin(X) \approx \frac{12.5567}{14.2} \][/tex]
[tex]\[ \sin(X) \approx 0.8843 \][/tex]
To get angle ∠X:
[tex]\[ X = \arcsin(0.8843) \][/tex]
[tex]\[ X \approx 62.3° \][/tex]
Therefore, the measure of angle ∠ZX (or mZZ, as denoted) is approximately:
[tex]\[ mZZ \approx 62.3° \][/tex]
So, the measure of mZZ to the nearest tenth is:
[tex]\[ \boxed{62.3°} \][/tex]
