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′(2,2)
, 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′(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′(2,6)
, 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′(2,8)
, Y′(10,0)
, Z′(12,4)
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′(2,2)
, 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′(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′(2,6)
, 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′(2,8)
, Y′(10,0)
, Z′(12,4)
Answers
GPT-4o mini
To find the coordinates of the dilated triangle \( X'Y'Z' \) when triangle \( XYZ \) is dilated by a factor of 2 with a center of dilation at the point \( (2,0) \), we can use the dilation formula:
\[
X' = (1 - k) \cdot C + k \cdot P
\]
where:
- \( k \) is the dilation factor (in this case, \( k = 2 \)),
- \( C \) is the center of dilation \( (2, 0) \),
- \( P \) is the original point.
For each vertex of triangle \( XYZ \), we need to find its coordinates under the dilation transformation. Let's denote the original points as:
- \( X = (x_1, y_1) \)
- \( Y = (x_2, y_2) \)
- \( Z = (x_3, y_3) \)
The coordinates of the dilated points \( X', Y', Z' \) will be calculated as follows:
\[
X' = (1 - 2)(2) + 2(x_1) = -2(2) + 2x_1 = 2x_1 - 4
\]
\[
Y' = (1 - 2)(2) + 2(x_2) = -2(2) + 2x_2 = 2x_2 - 4
\]
\[
Z' = (1 - 2)(2) + 2(x_3) = -2(2) + 2x_3 = 2x_3 - 4
\]
Since \( Y \) (and presumably the other points) have both \( x \) and \( y \) coordinates, we need the origin points.
Let's summarize:
1. Find the coordinates of the original triangle \( XYZ \).
2. Apply the dilation transformation based on the calculations above.
However, since we don't have the coordinates of triangle \( XYZ \) provided directly, we can't compute the exact values for \( X', Y', Z' \). But we can analyze the choices given in the question.
Based on the options provided, we can observe that each option shows a different configuration of dilated points. If you provide the coordinates of triangle \( XYZ \), I can help calculate the correct dilated coordinates accordingly. Without that, we can only analyze which transformations match with the described dilation centered at \( (2,0) \) and a factor of 2.
\[
X' = (1 - k) \cdot C + k \cdot P
\]
where:
- \( k \) is the dilation factor (in this case, \( k = 2 \)),
- \( C \) is the center of dilation \( (2, 0) \),
- \( P \) is the original point.
For each vertex of triangle \( XYZ \), we need to find its coordinates under the dilation transformation. Let's denote the original points as:
- \( X = (x_1, y_1) \)
- \( Y = (x_2, y_2) \)
- \( Z = (x_3, y_3) \)
The coordinates of the dilated points \( X', Y', Z' \) will be calculated as follows:
\[
X' = (1 - 2)(2) + 2(x_1) = -2(2) + 2x_1 = 2x_1 - 4
\]
\[
Y' = (1 - 2)(2) + 2(x_2) = -2(2) + 2x_2 = 2x_2 - 4
\]
\[
Z' = (1 - 2)(2) + 2(x_3) = -2(2) + 2x_3 = 2x_3 - 4
\]
Since \( Y \) (and presumably the other points) have both \( x \) and \( y \) coordinates, we need the origin points.
Let's summarize:
1. Find the coordinates of the original triangle \( XYZ \).
2. Apply the dilation transformation based on the calculations above.
However, since we don't have the coordinates of triangle \( XYZ \) provided directly, we can't compute the exact values for \( X', Y', Z' \). But we can analyze the choices given in the question.
Based on the options provided, we can observe that each option shows a different configuration of dilated points. If you provide the coordinates of triangle \( XYZ \), I can help calculate the correct dilated coordinates accordingly. Without that, we can only analyze which transformations match with the described dilation centered at \( (2,0) \) and a factor of 2.