Question
Find point H to prove that DH¯¯¯¯¯¯¯¯¯ is an enlargement of DF¯¯¯¯¯¯¯¯ by a scale factor of 2 with point D as the center of dilation.(1 point)ResponsesH is at (−2,0).upper H is at left parenthesis negative 2 comma 0 right parenthesis .H is at (8,18).upper H is at left parenthesis 8 comma 18 right parenthesis .H is at (5.5,3).upper H is at left parenthesis 5.5 comma 3 right parenthesis .H is at (13,6).
Answers
Answered by
GPT-4o mini
To find point \( H \) such that \( DH \) is an enlargement of \( DF \) by a scale factor of 2 with point \( D \) as the center of dilation, we need to apply the properties of dilation.
The formula for dilating a point \( F(x_F, y_F) \) from a center of dilation \( D(x_D, y_D) \) with a scale factor \( k \) is as follows:
\[
H = D + k(F - D)
\]
Where \( H \) is the dilated point, \( D \) is the center of dilation, \( F \) is the original point, and \( k \) is the scale factor.
Here, we set \( k = 2 \), so point \( H \) will be two times as far from \( D \) in the same direction as \( F \) is from \( D \).
To calculate the dilated point \( H \), you would need to know the coordinates of point \( D \) and point \( F \). However, the problem does not provide the coordinates for \( D \) or \( F \).
You can proceed by assuming that \( D \) is at the origin \((0, 0)\) for simplicity, then we will indicate that the position of \( H\) is determined based on \( F\):
If we have point \( F(x_F, y_F) \):
\[
H_x = 0 + 2(x_F - 0)
\]
\[
H_y = 0 + 2(y_F - 0)
\]
\[
H = (2x_F, 2y_F)
\]
If you provide a specific location for \( F \) or \( D \), I can compute \( H \) explicitly.
Assuming possible options, we can check which one can be the result of dilation given a point for \( F \).
Given the options:
1. \( H(-2, 0) \)
2. \( H(8, 18) \)
3. \( H(5.5, 3) \)
4. \( H(13, 6) \)
Suppose one possible point \( F \) was at \( (4, 9) \), for example.
\[
H = (2 \cdot 4, 2 \cdot 9) = (8, 18)
\]
In this case, \( H(8, 18) \) would satisfy this requirement of being an enlargement of \( DF \) (assuming suitable values for \( D \) and \( F \) based on context).
Thus, without additional details about \( D \) and \( F \), the most consistent answer with basic dilation principles would be:
**H is at (8, 18).**
The formula for dilating a point \( F(x_F, y_F) \) from a center of dilation \( D(x_D, y_D) \) with a scale factor \( k \) is as follows:
\[
H = D + k(F - D)
\]
Where \( H \) is the dilated point, \( D \) is the center of dilation, \( F \) is the original point, and \( k \) is the scale factor.
Here, we set \( k = 2 \), so point \( H \) will be two times as far from \( D \) in the same direction as \( F \) is from \( D \).
To calculate the dilated point \( H \), you would need to know the coordinates of point \( D \) and point \( F \). However, the problem does not provide the coordinates for \( D \) or \( F \).
You can proceed by assuming that \( D \) is at the origin \((0, 0)\) for simplicity, then we will indicate that the position of \( H\) is determined based on \( F\):
If we have point \( F(x_F, y_F) \):
\[
H_x = 0 + 2(x_F - 0)
\]
\[
H_y = 0 + 2(y_F - 0)
\]
\[
H = (2x_F, 2y_F)
\]
If you provide a specific location for \( F \) or \( D \), I can compute \( H \) explicitly.
Assuming possible options, we can check which one can be the result of dilation given a point for \( F \).
Given the options:
1. \( H(-2, 0) \)
2. \( H(8, 18) \)
3. \( H(5.5, 3) \)
4. \( H(13, 6) \)
Suppose one possible point \( F \) was at \( (4, 9) \), for example.
\[
H = (2 \cdot 4, 2 \cdot 9) = (8, 18)
\]
In this case, \( H(8, 18) \) would satisfy this requirement of being an enlargement of \( DF \) (assuming suitable values for \( D \) and \( F \) based on context).
Thus, without additional details about \( D \) and \( F \), the most consistent answer with basic dilation principles would be:
**H is at (8, 18).**