Question
The parallelogram E'F'G'H' is a dilation of the parallelogram EFGH. What is the scale factor of the dilation?
E= 1,2
F= -1,1
G= -1,-2
H= 1,-1
E= 1,2
F= -1,1
G= -1,-2
H= 1,-1
Answers
Answered by
GPT-4o mini
To find the scale factor of the dilation that maps the parallelogram EFGH to the parallelogram E'F'G'H', we need the coordinates of points E', F', G', and H'. However, since those coordinates are not provided in your question, I can still explain how to calculate the scale factor once you have those coordinates.
1. **Find the lengths of the sides of the original parallelogram EFGH.**
- Calculate the distance between points E and F, F and G, G and H, and H and E using the distance formula:
\[
\text{Distance} = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}
\]
- For the points E(1, 2), F(-1, 1), G(-1, -2), and H(1, -1):
- Length EF:
\[
EF = \sqrt{((-1) - 1)^2 + (1 - 2)^2} = \sqrt{(-2)^2 + (-1)^2} = \sqrt{4 + 1} = \sqrt{5}
\]
- Length FG:
\[
FG = \sqrt{((-1) - (-1))^2 + (-2 - 1)^2} = \sqrt{0^2 + (-3)^2} = \sqrt{9} = 3
\]
- Length GH:
\[
GH = \sqrt{(1 - (-1))^2 + (-1 - (-2))^2} = \sqrt{(2)^2 + (1)^2} = \sqrt{4 + 1} = \sqrt{5}
\]
- Length HE:
\[
HE = \sqrt{(1 - 1)^2 + (2 - (-1))^2} = \sqrt{0^2 + (3)^2} = 3
\]
2. **Determine the lengths of the corresponding sides in E'F'G'H'.**
- Similarly calculate the lengths of the sides for the points E', F', G', and H'.
3. **Calculate the scale factor.**
- The scale factor \( k \) of the dilation can be determined by comparing the lengths of corresponding sides:
\[
k = \frac{\text{Length of a side in E'F'G'H'}}{\text{Length of the corresponding side in EFGH}}
\]
You should take the ratio of any corresponding side lengths. If you obtain the same ratio for all corresponding sides, it confirms your dilation is consistent and gives the scale factor.
4. **Make sure that your points E', F', G', and H' are available.** Once you obtain those coordinates, you apply the above calculations to find your specific scale factor.
1. **Find the lengths of the sides of the original parallelogram EFGH.**
- Calculate the distance between points E and F, F and G, G and H, and H and E using the distance formula:
\[
\text{Distance} = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}
\]
- For the points E(1, 2), F(-1, 1), G(-1, -2), and H(1, -1):
- Length EF:
\[
EF = \sqrt{((-1) - 1)^2 + (1 - 2)^2} = \sqrt{(-2)^2 + (-1)^2} = \sqrt{4 + 1} = \sqrt{5}
\]
- Length FG:
\[
FG = \sqrt{((-1) - (-1))^2 + (-2 - 1)^2} = \sqrt{0^2 + (-3)^2} = \sqrt{9} = 3
\]
- Length GH:
\[
GH = \sqrt{(1 - (-1))^2 + (-1 - (-2))^2} = \sqrt{(2)^2 + (1)^2} = \sqrt{4 + 1} = \sqrt{5}
\]
- Length HE:
\[
HE = \sqrt{(1 - 1)^2 + (2 - (-1))^2} = \sqrt{0^2 + (3)^2} = 3
\]
2. **Determine the lengths of the corresponding sides in E'F'G'H'.**
- Similarly calculate the lengths of the sides for the points E', F', G', and H'.
3. **Calculate the scale factor.**
- The scale factor \( k \) of the dilation can be determined by comparing the lengths of corresponding sides:
\[
k = \frac{\text{Length of a side in E'F'G'H'}}{\text{Length of the corresponding side in EFGH}}
\]
You should take the ratio of any corresponding side lengths. If you obtain the same ratio for all corresponding sides, it confirms your dilation is consistent and gives the scale factor.
4. **Make sure that your points E', F', G', and H' are available.** Once you obtain those coordinates, you apply the above calculations to find your specific scale factor.