To find the measure of angle [tex]$Z$[/tex] in triangle [tex]$XYZ$[/tex] with vertices [tex]$X(-1,-1)$[/tex], [tex]$Y(-2,1)$[/tex], and [tex]$Z(1,2)$[/tex], we can follow these steps:
1. Calculate the distances (side lengths) between the points to determine the sides of the triangle:
- [tex]\( XY \)[/tex]: the distance between [tex]$X(-1,-1)$[/tex] and [tex]$Y(-2,1)$[/tex]
- [tex]\( YZ \)[/tex]: the distance between [tex]$Y(-2,1)$[/tex] and [tex]$Z(1,2)$[/tex]
- [tex]\( ZX \)[/tex]: the distance between [tex]$Z(1,2)$[/tex] and [tex]$X(-1,-1)$[/tex]
Using the distance formula:
[tex]\[
d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}
\][/tex]
Let's calculate each:
- [tex]\( XY = \sqrt{((-2) - (-1))^2 + (1 - (-1))^2} = \sqrt{(-1)^2 + (2)^2} = \sqrt{1 + 4} = 2.2361 \)[/tex]
- [tex]\( YZ = \sqrt{(1 - (-2))^2 + (2 - 1)^2} = \sqrt{(3)^2 + (1)^2} = \sqrt{9 + 1} = 3.1623 \)[/tex]
- [tex]\( ZX = \sqrt{(1 - (-1))^2 + (2 - (-1))^2} = \sqrt{(2)^2 + (3)^2} = \sqrt{4 + 9} = 3.6056 \)[/tex]
2. Use the Law of Cosines to find the measure of angle [tex]$\angle Z$[/tex]:
The Law of Cosines states:
[tex]\[
c^2 = a^2 + b^2 - 2ab \cdot \cos(C)
\][/tex]
In our case, [tex]\( c = YZ \)[/tex], [tex]\( a = XY \)[/tex], and [tex]\( b = ZX \)[/tex]. We want to solve for [tex]\(\cos Z\)[/tex]:
[tex]\[
\cos Z = \frac{a^2 + b^2 - c^2}{2ab}
\][/tex]
Substituting the calculated side lengths:
[tex]\[
\cos Z = \frac{XY^2 + ZX^2 - YZ^2}{2 \cdot XY \cdot ZX}
\][/tex]
Plugging in the values:
[tex]\[
\cos Z = \frac{(2.2361)^2 + (3.6056)^2 - (3.1623)^2}{2 \cdot 2.2361 \cdot 3.6056}
\][/tex]
[tex]\[
\cos Z = \frac{5 + 13 - 10}{16}
\][/tex]
[tex]\[
\cos Z \approx 0.4961
\][/tex]
3. Calculate angle [tex]$Z$[/tex] from the cosine value:
Knowing [tex]\(\cos Z \approx 0.4961\)[/tex], we use the inverse cosine function to find [tex]\(\angle Z\)[/tex]:
[tex]\[
\angle Z = \cos^{-1}(0.4961) \approx 1.0517 \text{ radians}
\][/tex]
4. Convert the angle from radians to degrees:
[tex]\[
\angle Z \text{ in degrees} = 1.0517 \text{ radians} \times \left(\frac{180^\circ}{\pi}\right) \approx 60.2551^\circ
\][/tex]
Thus, the approximate measure of angle [tex]\(Z\)[/tex] is [tex]\(60.2551^\circ\)[/tex].
This means the closest option is:
[tex]\[ \boxed{61.5^\circ} \][/tex]
