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 1 1 2 2 3 3 1/2
Answers
GPT-4o mini
To determine the magnitude of dilation from polygon ABCD to polygon A′B′C′D′ with respect to the origin, you need to compare the coordinates of corresponding vertices in both polygons.
The magnitude of the dilation can be calculated using the formula:
\[
\text{Magnitude of dilation} = \frac{\text{distance from the origin to a vertex of the image}}{\text{distance from the origin to the corresponding vertex of the pre-image}}
\]
This involves measuring the distance from the origin (0,0) to a vertex \(A\) of polygon ABCD and its corresponding vertex \(A'\) of polygon A′B′C′.
For example, if the coordinate of point A is (x, y) and the coordinate of point A' is (kx, ky), the dilation factor \(k\) can be found as:
\[
k = \frac{\sqrt{(kx)^2 + (ky)^2}}{\sqrt{x^2 + y^2}} = \frac{|k| \cdot \sqrt{x^2 + y^2}}{\sqrt{x^2 + y^2}} = |k|
\]
You would do this for one or more pairs of corresponding points to find the dilation constant.
Since the actual coordinates are not provided in your question, I cannot calculate the exact magnitude. However, you could apply this method using the coordinates of your specific polygons. Based on common variables of dilations observed, the options given (1, 2, 3, 1/2) suggest the dilation factor might be a simple fractional or whole number.
If you have specific coordinates for the points, please provide them and I can give a more exact answer.
The magnitude of the dilation can be calculated using the formula:
\[
\text{Magnitude of dilation} = \frac{\text{distance from the origin to a vertex of the image}}{\text{distance from the origin to the corresponding vertex of the pre-image}}
\]
This involves measuring the distance from the origin (0,0) to a vertex \(A\) of polygon ABCD and its corresponding vertex \(A'\) of polygon A′B′C′.
For example, if the coordinate of point A is (x, y) and the coordinate of point A' is (kx, ky), the dilation factor \(k\) can be found as:
\[
k = \frac{\sqrt{(kx)^2 + (ky)^2}}{\sqrt{x^2 + y^2}} = \frac{|k| \cdot \sqrt{x^2 + y^2}}{\sqrt{x^2 + y^2}} = |k|
\]
You would do this for one or more pairs of corresponding points to find the dilation constant.
Since the actual coordinates are not provided in your question, I cannot calculate the exact magnitude. However, you could apply this method using the coordinates of your specific polygons. Based on common variables of dilations observed, the options given (1, 2, 3, 1/2) suggest the dilation factor might be a simple fractional or whole number.
If you have specific coordinates for the points, please provide them and I can give a more exact answer.