1.The coordinates of rectangle ABCD are A(2, 2), B(2, 5), C(6, 5), and D(6, 2). Using rectangle ABCD, find the coordinates of the rectangle that results when the center of dilation is the origin and ABCD is dilated by a scale factor of 3.
the corners are now three times farther from the origin
just multiply each coordinate by 3
To find the coordinates of the rectangle that results from dilating rectangle ABCD by a scale factor of 3 with the center of dilation at the origin, we need to multiply the coordinates of each vertex of ABCD by the scale factor.
Let's go through the process step by step:
1. Find the center point of rectangle ABCD:
- The x-coordinate of the center is the average of the x-coordinates of the vertices: (2 + 2 + 6 + 6) / 4 = 4
- The y-coordinate of the center is the average of the y-coordinates of the vertices: (2 + 5 + 5 + 2) / 4 = 3.5
So, the center point of ABCD is (4, 3.5).
2. Apply the dilation to each vertex:
- For vertex A(2, 2):
- Multiply the x-coordinate by the scale factor: 2 * 3 = 6
- Multiply the y-coordinate by the scale factor: 2 * 3 = 6
The new coordinates of A' are (6, 6).
- For vertex B(2, 5):
- Multiply the x-coordinate by the scale factor: 2 * 3 = 6
- Multiply the y-coordinate by the scale factor: 5 * 3 = 15
The new coordinates of B' are (6, 15).
- For vertex C(6, 5):
- Multiply the x-coordinate by the scale factor: 6 * 3 = 18
- Multiply the y-coordinate by the scale factor: 5 * 3 = 15
The new coordinates of C' are (18, 15).
- For vertex D(6, 2):
- Multiply the x-coordinate by the scale factor: 6 * 3 = 18
- Multiply the y-coordinate by the scale factor: 2 * 3 = 6
The new coordinates of D' are (18, 6).
Therefore, the coordinates of the resulting rectangle after dilating rectangle ABCD by a scale factor of 3 with the center of dilation at the origin are:
A'(6, 6), B'(6, 15), C'(18, 15), and D'(18, 6).