Question
Find the coordinates of point H on AB¯¯¯¯¯¯¯¯ such that the ratio of AH to HB is 1:4. (1 point) Responses (0.75,1.5) left parenthesis Start Fraction 3 over 4 End Fraction comma Start Fraction 3 over 2 End Fraction right parenthesis (0.6,1.2) left parenthesis Start Fraction 3 over 5 End Fraction comma Start Fraction 6 over 5 End Fraction right parenthesis (−1,−2) left parenthesis negative 1 comma negative 2 right parenthesis (0.2,0.2) left parenthesis Start Fraction 1 over 5 End Fraction comma Start Fraction 1 over 5 End Fraction right parenthesis
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Answer
What are the coordinates of point X if it is located 23 of the way from A to B ? (1 point) Responses X(4,83) upper X left parenthesis 4 comma Start Fraction 8 over 3 Endfraction right parenthesis X(163,113) upper X left parenthesis 6 comma Start Fraction 11 over 3 End Fraction right parenthesis X(83,10) upper X left parenthesis Start Fraction 8 over 3 End Fraction comma 10 right parenthesis X(263,5)
Answered by
GPT-4o mini
To find the coordinates of point H on line segment AB such that the ratio \( \frac{AH}{HB} = \frac{1}{4} \), we can set up a parameterization based on the coordinates of points A and B. However, the coordinates of points A and B are not given in the problem statement. Assuming A and B have coordinates \( A(x_1, y_1) \) and \( B(x_2, y_2) \), we can proceed with the following approach.
Let the coordinates of point H be expressed in terms of A and B. The coordinates of point H can be calculated using the section formula or weighted average.
Since the ratio \( AH:HB = 1:4 \), we can express the position of H as follows:
\[ H = \left( \frac{4x_1 + 1x_2}{4 + 1}, \frac{4y_1 + 1y_2}{4 + 1} \right) \]
This means H divides the segment AB into 1 part from A and 4 parts towards B.
The next step would require substituting the actual coordinates of points A and B to find H. However, without specific values for A and B, we cannot calculate the exact coordinates for H.
If we assume generic coordinates for A and B, we can check each provided response by substituting them back to see if they maintain the ratio \( \frac{1}{4} \).
For example:
1. For point \( H (0.75, 1.5) \)
2. For point \( H \left( \frac{3}{4}, \frac{3}{2} \right) \)
3. For point \( H (0.6, 1.2) \)
4. For point \( H \left( \frac{3}{5}, \frac{6}{5} \right) \)
5. For point \( H (-1, -2) \)
6. For point \( H (0.2, 0.2) \)
Each one must be validated against appropriate points A and B.
Since no values for A and B were provided, it is not possible to determine a definitive answer for H without additional information about the coordinates of A and B.
Let the coordinates of point H be expressed in terms of A and B. The coordinates of point H can be calculated using the section formula or weighted average.
Since the ratio \( AH:HB = 1:4 \), we can express the position of H as follows:
\[ H = \left( \frac{4x_1 + 1x_2}{4 + 1}, \frac{4y_1 + 1y_2}{4 + 1} \right) \]
This means H divides the segment AB into 1 part from A and 4 parts towards B.
The next step would require substituting the actual coordinates of points A and B to find H. However, without specific values for A and B, we cannot calculate the exact coordinates for H.
If we assume generic coordinates for A and B, we can check each provided response by substituting them back to see if they maintain the ratio \( \frac{1}{4} \).
For example:
1. For point \( H (0.75, 1.5) \)
2. For point \( H \left( \frac{3}{4}, \frac{3}{2} \right) \)
3. For point \( H (0.6, 1.2) \)
4. For point \( H \left( \frac{3}{5}, \frac{6}{5} \right) \)
5. For point \( H (-1, -2) \)
6. For point \( H (0.2, 0.2) \)
Each one must be validated against appropriate points A and B.
Since no values for A and B were provided, it is not possible to determine a definitive answer for H without additional information about the coordinates of A and B.
Answered by
GPT-4o mini
To find the coordinates of point \( X \) located \( \frac{2}{3} \) of the way from point \( A \) to point \( B \), we again need the coordinates of points \( A \) and \( B \). However, without specific coordinates for \( A \) and \( B \), we can describe how to find the coordinates of point \( X \).
Assuming the coordinates of points \( A \) and \( B \) are \( A(x_1, y_1) \) and \( B(x_2, y_2) \), the coordinates of point \( X \) can be calculated using the following formula:
\[
X = \left( x_1 + \frac{2}{3}(x_2 - x_1), \, y_1 + \frac{2}{3}(y_2 - y_1) \right)
\]
Now let's evaluate the given response options:
1. \( X(4, 83) \)
2. \( X \left( 4, \frac{8}{3} \right) \)
3. \( X(163, 113) \)
4. \( X \left( 6, \frac{11}{3} \right) \)
5. \( X(83, 10) \)
6. \( X \left( \frac{8}{3}, 10 \right) \)
7. \( X(263, 5) \)
Without the actual coordinates of points \( A \) and \( B \), we cannot determine the correct coordinates of point \( X \) directly. However, if you provide the coordinates of points \( A \) and \( B \), I will be able to calculate the coordinates of point \( X \) accurately or validate one of the provided options.
Assuming the coordinates of points \( A \) and \( B \) are \( A(x_1, y_1) \) and \( B(x_2, y_2) \), the coordinates of point \( X \) can be calculated using the following formula:
\[
X = \left( x_1 + \frac{2}{3}(x_2 - x_1), \, y_1 + \frac{2}{3}(y_2 - y_1) \right)
\]
Now let's evaluate the given response options:
1. \( X(4, 83) \)
2. \( X \left( 4, \frac{8}{3} \right) \)
3. \( X(163, 113) \)
4. \( X \left( 6, \frac{11}{3} \right) \)
5. \( X(83, 10) \)
6. \( X \left( \frac{8}{3}, 10 \right) \)
7. \( X(263, 5) \)
Without the actual coordinates of points \( A \) and \( B \), we cannot determine the correct coordinates of point \( X \) directly. However, if you provide the coordinates of points \( A \) and \( B \), I will be able to calculate the coordinates of point \( X \) accurately or validate one of the provided options.
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