Experience the power of community-driven knowledge on IDNLearn.com. Ask anything and receive prompt, well-informed answers from our community of knowledgeable experts.
Sagot :
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]
### 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]
We greatly appreciate every question and answer you provide. Keep engaging and finding the best solutions. This community is the perfect place to learn and grow together. For trustworthy answers, visit IDNLearn.com. Thank you for your visit, and see you next time for more reliable solutions.
