Question
In the image, polygon ABCD is dilated to form polygon A′B′C′D′ with respect to the origin. What is the magnitude of dilation? (1 point) Responses 12 Start Fraction 1 over 2 End Fraction 2 2 1 1 3 3 Skip to navigation. A -1,-1 A -2,-2 B -1,1 B -2,2 C 4,6 C 2,3 D 2, -1 D 4,-2
Answers
GPT-4o mini
To find the magnitude of the dilation from polygon ABCD to polygon A'B'C'D', we can determine the scale factor based on the coordinates of corresponding points in the two polygons.
We will calculate the scale factor using the coordinates provided:
1. Identify the coordinates of points A, B, C, and D for polygon ABCD and the corresponding points A', B', C', and D' for polygon A'B'C'D'.
From your response, we have:
- A (-1, -1) → A' (-2, -2)
- B (-1, 1) → B' (-2, 2)
- C (4, 6) → C' (2, 3)
- D (2, -1) → D' (4, -2)
2. The scale factor \( k \) for dilation from point A to A' can be calculated by the formula:
\[
k = \frac{\text{Coordinate of A'}}{\text{Coordinate of A}}
\]
3. We will calculate \( k \) for each of the points:
For A to A':
\[
k_A = \frac{-2}{-1} = 2
\]
For B to B':
\[
k_B = \frac{-2}{-1} = 2
\]
(Note that y-coordinates need to be checked as well.)
Since B's y-coordinate is 1 and goes to 2, it should also be checked:
\[
k_B (y) = \frac{2}{1} = 2
\]
For C to C':
\[
k_C = \frac{2}{4} = \frac{1}{2}
\]
For D to D':
\[
k_D = \frac{4}{2} = 2
\]
4. Since the scale from A, B, and D gives us a consistent scale factor of \( k = 2\) and for C a different one of \( \frac{1}{2}\), it appears there might be inconsistencies which may mean polygons ABCD and A'B'C'D' do not conform consistently to a single dilation.
However, the majority of points A, B, and D suggest a scale factor of \( 2\).
So the most likely correct answer for the dilation magnitude (assuming a consistent proportion exists among most points) would be:
**2**.
We will calculate the scale factor using the coordinates provided:
1. Identify the coordinates of points A, B, C, and D for polygon ABCD and the corresponding points A', B', C', and D' for polygon A'B'C'D'.
From your response, we have:
- A (-1, -1) → A' (-2, -2)
- B (-1, 1) → B' (-2, 2)
- C (4, 6) → C' (2, 3)
- D (2, -1) → D' (4, -2)
2. The scale factor \( k \) for dilation from point A to A' can be calculated by the formula:
\[
k = \frac{\text{Coordinate of A'}}{\text{Coordinate of A}}
\]
3. We will calculate \( k \) for each of the points:
For A to A':
\[
k_A = \frac{-2}{-1} = 2
\]
For B to B':
\[
k_B = \frac{-2}{-1} = 2
\]
(Note that y-coordinates need to be checked as well.)
Since B's y-coordinate is 1 and goes to 2, it should also be checked:
\[
k_B (y) = \frac{2}{1} = 2
\]
For C to C':
\[
k_C = \frac{2}{4} = \frac{1}{2}
\]
For D to D':
\[
k_D = \frac{4}{2} = 2
\]
4. Since the scale from A, B, and D gives us a consistent scale factor of \( k = 2\) and for C a different one of \( \frac{1}{2}\), it appears there might be inconsistencies which may mean polygons ABCD and A'B'C'D' do not conform consistently to a single dilation.
However, the majority of points A, B, and D suggest a scale factor of \( 2\).
So the most likely correct answer for the dilation magnitude (assuming a consistent proportion exists among most points) would be:
**2**.