Question
Parallelogram ABCD, shown on the coordinate grid below, is dilated by a scale factor 4, centered at the origin. What are the vertex coordinates of the dilated figure?
image.png
A'(4,-12), B`(16,4), C'(12,16), D'(0,0)
A'(12,-4), B`(-4,-16), C'(-16,-12), D'(0,0)
A'(-12,4), B`(4,16), C'(16,12), D'(0,0)
A'(-4,12), B`(-16,-4), C'(-12,-16), D'(0,0)
image.png
A'(4,-12), B`(16,4), C'(12,16), D'(0,0)
A'(12,-4), B`(-4,-16), C'(-16,-12), D'(0,0)
A'(-12,4), B`(4,16), C'(16,12), D'(0,0)
A'(-4,12), B`(-16,-4), C'(-12,-16), D'(0,0)
Answers
Answered by
GPT-4o mini
To determine the vertex coordinates of the dilated figure, we need to apply the dilation transformation to each vertex of the parallelogram ABCD using the scale factor of 4 and the center of dilation at the origin (0, 0).
The formula for dilation centered at the origin is:
\[
(x', y') = (kx, ky)
\]
where \((x, y)\) are the original coordinates of a point, \((x', y')\) are the new coordinates after dilation, and \(k\) is the scale factor.
Let's denote the original coordinates of the vertices of parallelogram ABCD as \(A(x_1, y_1)\), \(B(x_2, y_2)\), \(C(x_3, y_3)\), and \(D(x_4, y_4)\).
Since we don't have the specific coordinates of the original parallelogram ABCD from your description, I will show you how to calculate the coordinates of the dilated vertices. For each vertex:
1. Multiply the x-coordinate of the vertex by the scale factor (4).
2. Multiply the y-coordinate of the vertex by the scale factor (4).
For example, if vertex A has coordinates \( (x_1, y_1) \):
\[
A' = (4 \cdot x_1, 4 \cdot y_1)
\]
Since I don't have the original coordinates, let’s assume possible original vertices for analysis:
1. Let's suppose we have:
- A(3, -3)
- B(4, 1)
- C(3, 4)
- D(0, 0)
Using our method:
\[
A' = (4 \cdot 3, 4 \cdot -3) = (12, -12)
\]
\[
B' = (4 \cdot 4, 4 \cdot 1) = (16, 4)
\]
\[
C' = (4 \cdot 3, 4 \cdot 4) = (12, 16)
\]
\[
D' = (4 \cdot 0, 4 \cdot 0) = (0, 0)
\]
Thus, if the original vertices were indeed A(3, -3), B(4, 1), C(3, 4), D(0, 0), our final figures after dilation would be:
A'(12, -12), B'(16, 4), C'(12, 16), D'(0, 0).
Referring to your options, the set of coordinates we calculated (without knowing the original coordinates of ABCD) suggests that there might be a matching pattern.
However, since I need the original coordinates to give the final output, please provide the coordinates of the original vertices for an accurate dilation or check which of the given answers aligns with what you expect based on the dilated scale of factor 4.
The formula for dilation centered at the origin is:
\[
(x', y') = (kx, ky)
\]
where \((x, y)\) are the original coordinates of a point, \((x', y')\) are the new coordinates after dilation, and \(k\) is the scale factor.
Let's denote the original coordinates of the vertices of parallelogram ABCD as \(A(x_1, y_1)\), \(B(x_2, y_2)\), \(C(x_3, y_3)\), and \(D(x_4, y_4)\).
Since we don't have the specific coordinates of the original parallelogram ABCD from your description, I will show you how to calculate the coordinates of the dilated vertices. For each vertex:
1. Multiply the x-coordinate of the vertex by the scale factor (4).
2. Multiply the y-coordinate of the vertex by the scale factor (4).
For example, if vertex A has coordinates \( (x_1, y_1) \):
\[
A' = (4 \cdot x_1, 4 \cdot y_1)
\]
Since I don't have the original coordinates, let’s assume possible original vertices for analysis:
1. Let's suppose we have:
- A(3, -3)
- B(4, 1)
- C(3, 4)
- D(0, 0)
Using our method:
\[
A' = (4 \cdot 3, 4 \cdot -3) = (12, -12)
\]
\[
B' = (4 \cdot 4, 4 \cdot 1) = (16, 4)
\]
\[
C' = (4 \cdot 3, 4 \cdot 4) = (12, 16)
\]
\[
D' = (4 \cdot 0, 4 \cdot 0) = (0, 0)
\]
Thus, if the original vertices were indeed A(3, -3), B(4, 1), C(3, 4), D(0, 0), our final figures after dilation would be:
A'(12, -12), B'(16, 4), C'(12, 16), D'(0, 0).
Referring to your options, the set of coordinates we calculated (without knowing the original coordinates of ABCD) suggests that there might be a matching pattern.
However, since I need the original coordinates to give the final output, please provide the coordinates of the original vertices for an accurate dilation or check which of the given answers aligns with what you expect based on the dilated scale of factor 4.