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) Responses H is at (8,18) . upper H is at left parenthesis 8 comma 18 right parenthesis . H is at (−2,0) . upper H is at left parenthesis negative 2 comma 0 right parenthesis . H is at (13,6) . upper H is at left parenthesis 13 comma 6 right parenthesis . H is at (5.5,3) .
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 can use the formula for dilation.
If point D has coordinates \( (x_D, y_D) \) and point F has coordinates \( (x_F, y_F) \), then the coordinates of point H, which is the dilated point, can be calculated using the formula:
\[
H_x = x_D + 2(x_F - x_D)
\]
\[
H_y = y_D + 2(y_F - y_D)
\]
This means that point H will be located at double the distance from D in the direction of F.
Let's assume point D is the origin, that is \( D = (0,0) \).
Given the possible options, we can calculate the position of H based on the given forms of DF and check which point lies at double the distance from D compared to F. Without the coordinates of F, we cannot explicitly calculate F and verify H.
However, if we assume we are verifying possible positions of H knowing point D and the hypothesis that \( DH \) is twice as long as \( DF \), we can check the given point options in relation to point D.
The given options are:
1. H is at (8, 18)
2. H is at (-2, 0)
3. H is at (13, 6)
4. H is at (5.5, 3)
### Testing the points:
1. For \( H = (8, 18) \): This point is not twice the distance from D compared with any reasonable point F around the origin without more context.
2. For \( H = (-2, 0) \): This point is clearly closer to D itself than any scaling of DF.
3. For \( H = (13, 6) \): This point lies outside the original triangle/segment DF as stated.
4. For \( H = (5.5, 3) \): Point H still does not seem to extend reasonably based on coordinates.
#### Summary:
To validate which is the proper answer among these choices, the relationship between D and H through a corresponding F must be established, or we need to know the actual point F's coordinates.
If you have the coordinates of point F, that would help determine the correct answer for H. Otherwise, without that information, none of the points unequivocally proves correct enlargement criterion.
Please supply coordinates for point F to establish context for the enlargement relative to point D.
If point D has coordinates \( (x_D, y_D) \) and point F has coordinates \( (x_F, y_F) \), then the coordinates of point H, which is the dilated point, can be calculated using the formula:
\[
H_x = x_D + 2(x_F - x_D)
\]
\[
H_y = y_D + 2(y_F - y_D)
\]
This means that point H will be located at double the distance from D in the direction of F.
Let's assume point D is the origin, that is \( D = (0,0) \).
Given the possible options, we can calculate the position of H based on the given forms of DF and check which point lies at double the distance from D compared to F. Without the coordinates of F, we cannot explicitly calculate F and verify H.
However, if we assume we are verifying possible positions of H knowing point D and the hypothesis that \( DH \) is twice as long as \( DF \), we can check the given point options in relation to point D.
The given options are:
1. H is at (8, 18)
2. H is at (-2, 0)
3. H is at (13, 6)
4. H is at (5.5, 3)
### Testing the points:
1. For \( H = (8, 18) \): This point is not twice the distance from D compared with any reasonable point F around the origin without more context.
2. For \( H = (-2, 0) \): This point is clearly closer to D itself than any scaling of DF.
3. For \( H = (13, 6) \): This point lies outside the original triangle/segment DF as stated.
4. For \( H = (5.5, 3) \): Point H still does not seem to extend reasonably based on coordinates.
#### Summary:
To validate which is the proper answer among these choices, the relationship between D and H through a corresponding F must be established, or we need to know the actual point F's coordinates.
If you have the coordinates of point F, that would help determine the correct answer for H. Otherwise, without that information, none of the points unequivocally proves correct enlargement criterion.
Please supply coordinates for point F to establish context for the enlargement relative to point D.