Question
If you dilate triangle XYZ to form X′Y′Z′ using a magnitude of 2 and a center of dilation at (2,0) , what are the coordinates of the dilated triangle? (1 point) Responses X′(2,6) , Y′(8,0) , Z′(9.5,3) upper X prime left parenthesis 8 comma 4 right parenthesis , upper Y prime left parenthesis 12 comma 0 right parenthesis , upper Z prime left parenthesis 13 comma 2 right parenthesis X′(6,4) , Y′(10,0) , Z′(11,4) upper X prime left parenthesis 4 comma 8 right parenthesis , upper Y prime left parenthesis 12 comma 0 right parenthesis , upper Z prime left parenthesis 14 comma 4 right parenthesis X′(2,2) , Y′(4,0) , Z′(4.5,1) upper X prime left parenthesis negative 4 comma 4 right parenthesis , upper Y prime left parenthesis 0 comma 0 right parenthesis , upper Z prime left parenthesis 1 comma 2 right parenthesis X′(2,8) , Y′(10,0) , Z′(12,4)
Answers
GPT-4o mini
To find the coordinates of the dilated triangle X′Y′Z′, we can use the formula for dilation. The dilation of a point (x, y) around a center of dilation (cx, cy) with a magnitude of k is given by the formula:
\[
(x', y') = (cx + k(x - cx), cy + k(y - cy))
\]
In this case, the center of dilation is (2, 0) and the magnitude is 2. Let's apply this to each point of triangle XYZ. We don't have the coordinates of triangle XYZ, so let's assume them as (x₁, y₁), (x₂, y₂), and (x₃, y₃) temporarily.
Using the dilation formula, we get:
- For point X (x₁, y₁):
\[
X' = (2 + 2(x₁ - 2), 0 + 2(y₁ - 0)) = (2 + 2x₁ - 4, 2y₁) = (2x₁ - 2, 2y₁)
\]
- For point Y (x₂, y₂):
\[
Y' = (2 + 2(x₂ - 2), 0 + 2(y₂ - 0)) = (2 + 2x₂ - 4, 2y₂) = (2x₂ - 2, 2y₂)
\]
- For point Z (x₃, y₃):
\[
Z' = (2 + 2(x₃ - 2), 0 + 2(y₃ - 0)) = (2 + 2x₃ - 4, 2y₃) = (2x₃ - 2, 2y₃)
\]
Now, without the original coordinates of triangle XYZ, we can't calculate the exact coordinates of X′, Y′, and Z′. However, we can examine the response options given in the problem.
1. **X′(2, 6)**, **Y′(8, 0)**, **Z′(9.5, 3)**
2. **X′(8, 4)**, **Y′(12, 0)**, **Z′(13, 2)**
3. **X′(6, 4)**, **Y′(10, 0)**, **Z′(11, 4)**
4. **X′(4, 8)**, **Y′(12, 0)**, **Z′(14, 4)**
5. **X′(2, 2)**, **Y′(4, 0)**, **Z′(4.5, 1)**
6. **X′(-4, 4)**, **Y′(0, 0)**, **Z′(1, 2)**
7. **X′(2, 8)**, **Y′(10, 0)**, **Z′(12, 4)**
Given that we need to rely on the dilation from the center (2, 0) with a magnitude of 2, the second set of values: **X′(8, 4)**, **Y′(12, 0)**, **Z′(13, 2)** seems plausible as the dilated points would be plausible for an original set of points that are closer to the center.
Thus, without the original coordinates, the best candidate based on the dilation transformations is:
**X′(8, 4)**, **Y′(12, 0)**, **Z′(13, 2)**.
\[
(x', y') = (cx + k(x - cx), cy + k(y - cy))
\]
In this case, the center of dilation is (2, 0) and the magnitude is 2. Let's apply this to each point of triangle XYZ. We don't have the coordinates of triangle XYZ, so let's assume them as (x₁, y₁), (x₂, y₂), and (x₃, y₃) temporarily.
Using the dilation formula, we get:
- For point X (x₁, y₁):
\[
X' = (2 + 2(x₁ - 2), 0 + 2(y₁ - 0)) = (2 + 2x₁ - 4, 2y₁) = (2x₁ - 2, 2y₁)
\]
- For point Y (x₂, y₂):
\[
Y' = (2 + 2(x₂ - 2), 0 + 2(y₂ - 0)) = (2 + 2x₂ - 4, 2y₂) = (2x₂ - 2, 2y₂)
\]
- For point Z (x₃, y₃):
\[
Z' = (2 + 2(x₃ - 2), 0 + 2(y₃ - 0)) = (2 + 2x₃ - 4, 2y₃) = (2x₃ - 2, 2y₃)
\]
Now, without the original coordinates of triangle XYZ, we can't calculate the exact coordinates of X′, Y′, and Z′. However, we can examine the response options given in the problem.
1. **X′(2, 6)**, **Y′(8, 0)**, **Z′(9.5, 3)**
2. **X′(8, 4)**, **Y′(12, 0)**, **Z′(13, 2)**
3. **X′(6, 4)**, **Y′(10, 0)**, **Z′(11, 4)**
4. **X′(4, 8)**, **Y′(12, 0)**, **Z′(14, 4)**
5. **X′(2, 2)**, **Y′(4, 0)**, **Z′(4.5, 1)**
6. **X′(-4, 4)**, **Y′(0, 0)**, **Z′(1, 2)**
7. **X′(2, 8)**, **Y′(10, 0)**, **Z′(12, 4)**
Given that we need to rely on the dilation from the center (2, 0) with a magnitude of 2, the second set of values: **X′(8, 4)**, **Y′(12, 0)**, **Z′(13, 2)** seems plausible as the dilated points would be plausible for an original set of points that are closer to the center.
Thus, without the original coordinates, the best candidate based on the dilation transformations is:
**X′(8, 4)**, **Y′(12, 0)**, **Z′(13, 2)**.