Question
The triangle T'U'V' is a dilation of the triangle TUV. What is the scale factor of the dilation?
T'(-3,3)
U'(0,3)
V'(-3,-9)
T(-1,1)
U(0,1)
V(-1,-3)
Simplify your answer and write it as a proper fraction, an improper fraction, or a whole number.
T'(-3,3)
U'(0,3)
V'(-3,-9)
T(-1,1)
U(0,1)
V(-1,-3)
Simplify your answer and write it as a proper fraction, an improper fraction, or a whole number.
Answers
GPT-4o mini
To find the scale factor of the dilation from triangle TUV to triangle T'U'V', we can compare the coordinates of corresponding points.
We will use the formula for the scale factor \( k \):
\[
k = \frac{\text{length of a side in the new triangle}}{\text{length of the corresponding side in the original triangle}}
\]
We first calculate the lengths of corresponding sides in both triangles.
1. **Calculating side lengths of triangle TUV**:
Using the distance formula:
\[
d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}
\]
- Length of side TU:
- \( T(-1, 1) \) and \( U(0, 1) \)
\[
TU = \sqrt{(0 - (-1))^2 + (1 - 1)^2} = \sqrt{1^2 + 0^2} = \sqrt{1} = 1
\]
- Length of side UV:
- \( U(0, 1) \) and \( V(-1, -3) \)
\[
UV = \sqrt{((-1) - 0)^2 + ((-3) - 1)^2} = \sqrt{(-1)^2 + (-4)^2} = \sqrt{1 + 16} = \sqrt{17}
\]
- Length of side TV:
- \( T(-1, 1) \) and \( V(-1, -3) \)
\[
TV = \sqrt{((-1) - (-1))^2 + ((-3) - 1)^2} = \sqrt{0^2 + (-4)^2} = \sqrt{16} = 4
\]
2. **Calculating side lengths of triangle T'U'V'**:
Using the same distance formula for triangle T'U'V':
- Length of side T'U':
- \( T'(-3, 3) \) and \( U'(0, 3) \)
\[
T'U' = \sqrt{(0 - (-3))^2 + (3 - 3)^2} = \sqrt{3^2 + 0^2} = \sqrt{9} = 3
\]
- Length of side U'V':
- \( U'(0, 3) \) and \( V'(-3, -9) \)
\[
U'V' = \sqrt{((-3) - 0)^2 + ((-9) - 3)^2} = \sqrt{(-3)^2 + (-12)^2} = \sqrt{9 + 144} = \sqrt{153}
\]
- Length of side T'V':
- \( T'(-3, 3) \) and \( V'(-3, -9) \)
\[
T'V' = \sqrt{((-3) - (-3))^2 + ((-9) - 3)^2} = \sqrt{0^2 + (-12)^2} = \sqrt{144} = 12
\]
3. **Finding the scale factor**:
We can calculate the scale factor \( k \) using the lengths we calculated:
- Using side \( TU \) and \( T'U' \):
\[
k = \frac{T'U'}{TU} = \frac{3}{1} = 3
\]
So, the scale factor of the dilation from triangle TUV to triangle T'U'V' is:
\[
\boxed{3}
\]
We will use the formula for the scale factor \( k \):
\[
k = \frac{\text{length of a side in the new triangle}}{\text{length of the corresponding side in the original triangle}}
\]
We first calculate the lengths of corresponding sides in both triangles.
1. **Calculating side lengths of triangle TUV**:
Using the distance formula:
\[
d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}
\]
- Length of side TU:
- \( T(-1, 1) \) and \( U(0, 1) \)
\[
TU = \sqrt{(0 - (-1))^2 + (1 - 1)^2} = \sqrt{1^2 + 0^2} = \sqrt{1} = 1
\]
- Length of side UV:
- \( U(0, 1) \) and \( V(-1, -3) \)
\[
UV = \sqrt{((-1) - 0)^2 + ((-3) - 1)^2} = \sqrt{(-1)^2 + (-4)^2} = \sqrt{1 + 16} = \sqrt{17}
\]
- Length of side TV:
- \( T(-1, 1) \) and \( V(-1, -3) \)
\[
TV = \sqrt{((-1) - (-1))^2 + ((-3) - 1)^2} = \sqrt{0^2 + (-4)^2} = \sqrt{16} = 4
\]
2. **Calculating side lengths of triangle T'U'V'**:
Using the same distance formula for triangle T'U'V':
- Length of side T'U':
- \( T'(-3, 3) \) and \( U'(0, 3) \)
\[
T'U' = \sqrt{(0 - (-3))^2 + (3 - 3)^2} = \sqrt{3^2 + 0^2} = \sqrt{9} = 3
\]
- Length of side U'V':
- \( U'(0, 3) \) and \( V'(-3, -9) \)
\[
U'V' = \sqrt{((-3) - 0)^2 + ((-9) - 3)^2} = \sqrt{(-3)^2 + (-12)^2} = \sqrt{9 + 144} = \sqrt{153}
\]
- Length of side T'V':
- \( T'(-3, 3) \) and \( V'(-3, -9) \)
\[
T'V' = \sqrt{((-3) - (-3))^2 + ((-9) - 3)^2} = \sqrt{0^2 + (-12)^2} = \sqrt{144} = 12
\]
3. **Finding the scale factor**:
We can calculate the scale factor \( k \) using the lengths we calculated:
- Using side \( TU \) and \( T'U' \):
\[
k = \frac{T'U'}{TU} = \frac{3}{1} = 3
\]
So, the scale factor of the dilation from triangle TUV to triangle T'U'V' is:
\[
\boxed{3}
\]