Sure, let's carefully determine the approximate measure of angle [tex]\( Z \)[/tex] in triangle [tex]\( XYZ \)[/tex], given its vertices [tex]\( X(-1, -1) \)[/tex], [tex]\( Y(-2, 1) \)[/tex], and [tex]\( Z(1, 2) \)[/tex].
### Step 1: Calculate the lengths of the sides
First, we calculate the lengths of the sides of triangle [tex]\( XYZ \)[/tex].
- Length [tex]\( a \)[/tex] (opposite to vertex [tex]\(Z\)[/tex]): this is the distance between points [tex]\( Y \)[/tex] and [tex]\( X \)[/tex].
- Length [tex]\( b \)[/tex] (opposite to vertex [tex]\( Y\)[/tex]): this is the distance between points [tex]\( X \)[/tex] and [tex]\( Z \)[/tex].
- Length [tex]\( c \)[/tex] (opposite to vertex [tex]\( X\)[/tex]): this is the distance between points [tex]\( Y \)[/tex] and [tex]\( Z \)[/tex].
1. Distance [tex]\( a = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \)[/tex]:
[tex]\[
a = \sqrt{(1 - (-2))^2 + (2 - 1)^2}
\][/tex]
[tex]\[
a = \sqrt{(1 + 2)^2 + (2 - 1)^2} = \sqrt{3^2 + 1^2} = \sqrt{9 + 1} = \sqrt{10} \approx 3.162
\][/tex]
2. Distance [tex]\( b = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \)[/tex]:
[tex]\[
b = \sqrt{(1 - (-1))^2 + (2 - (-1))^2}
\][/tex]
[tex]\[
b = \sqrt{(1 + 1)^2 + (2 + 1)^2} = \sqrt{2^2 + 3^2} = \sqrt{4 + 9} = \sqrt{13} \approx 3.606
\][/tex]
3. Distance [tex]\( c = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \)[/tex]:
[tex]\[
c = \sqrt{((-2) - (-1))^2 + (1 - (-1))^2}
\][/tex]
[tex]\[
c = \sqrt{(-2 + 1)^2 + (1 + 1)^2} = \sqrt{(-1)^2 + 2^2} = \sqrt{1 + 4} = \sqrt{5} \approx 2.236
\][/tex]
### Step 2: Calculate the measure of angle [tex]\( Z \)[/tex] using the Law of Cosines
The Law of Cosines states that:
[tex]\[
\cos(Z) = \frac{a^2 + b^2 - c^2}{2ab}
\][/tex]
Plugging in the values [tex]\( a \approx 3.162 \)[/tex], [tex]\( b \approx 3.606 \)[/tex], and [tex]\( c \approx 2.236 \)[/tex]:
[tex]\[
\cos(Z) = \frac{3.162^2 + 3.606^2 - 2.236^2}{2 \cdot 3.162 \cdot 3.606}
\][/tex]
[tex]\[
\cos(Z) = \frac{10 + 13 - 5}{2 \cdot 3.162 \cdot 3.606}
\][/tex]
[tex]\[
\cos(Z) = \frac{18}{22.8} \approx 0.789
\][/tex]
### Step 3: Find the angle in degrees
To find [tex]\( Z \)[/tex] in degrees, we take the inverse cosine and convert it to degrees:
[tex]\[
Z \approx \cos^{-1}(0.789) \approx 37.875^\circ
\][/tex]
### Step 4: Approximate the value to the nearest answer choice
Comparing [tex]\( 37.875^\circ \)[/tex] to the provided answer choices, the closest value is about [tex]\( 37.2^\circ \)[/tex].
Thus, the approximate measure of angle [tex]\( Z \)[/tex] is [tex]\( 37.2^\circ \)[/tex].
[tex]\(\boxed{37.2^\circ}\)[/tex]
