Question
Find the coordinates of point H on AB¯¯¯¯¯¯¯¯ such that the ratio of AH to HB is 1:4.(1 point)Responses(−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(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)
Answers
Answered by
GPT-4o mini
To find the coordinates of point \( H \) on line segment \( AB \) such that the ratio of \( AH \) to \( HB \) is \( 1:4 \), we can use the section formula.
Let \( A(x_1, y_1) \) and \( B(x_2, y_2) \) be the coordinates of points \( A \) and \( B \). The coordinates of a point \( H \) that divides \( AB \) in the ratio \( m:n \) can be found using the formula:
\[
H\left(\frac{mx_2 + nx_1}{m+n}, \frac{my_2 + ny_1}{m+n}\right)
\]
In this case, \( m = 1 \) (for segment \( AH \)) and \( n = 4 \) (for segment \( HB \)). Therefore, the ratio \( AH:HB \) is \( 1:4 \).
This means \( H \) divides \( AB \) into 5 equal parts, with \( AH \) being 1 part and \( HB \) being 4 parts.
Assuming the coordinates of \( A \) and \( B \) are as follows (since specific coordinates are not given):
- Let \( A = (x_1, y_1) \)
- Let \( B = (x_2, y_2) \)
Using the section formula:
\[
H\left(\frac{1 \cdot x_2 + 4 \cdot x_1}{1+4}, \frac{1 \cdot y_2 + 4 \cdot y_1}{1+4}\right)
\]
This simplifies to:
\[
H\left(\frac{x_2 + 4x_1}{5}, \frac{y_2 + 4y_1}{5}\right)
\]
Now, we compare \( H \) with the provided coordinate options to find which one corresponds to this calculation.
However, without specific coordinates for \( A \) and \( B \), we must select an appropriate choice from the provided options based purely on ratio consideration and the requirement that the point divides \( AB \) in a 1:4 ratio.
Given the choices:
1. \( (-1, -2) \)
2. \( \left(\frac{1}{5}, \frac{1}{5}\right) \)
3. \( \left(\frac{3}{4}, \frac{3}{2}\right) \)
4. \( (0.6, 1.2) \)
If we calculate for \( H \) to make it in \( 1:4 \) ratio from \( A \) to \( B \), the fractions suggest that \( H \) should be closer towards point \( A \) if \( A \) is lower than \( B \) on the coordinate scale; since \( (0.2, 0.2) \) keeps both values low, it’s a strong candidate.
Therefore, the most logical option given these constraints would be \( \left(\frac{1}{5}, \frac{1}{5}\right) \) which reflects the lower ratio and maintains the proportional sequence \( AH \) and \( HB \).
### Final Answer:
\[
\left(\frac{1}{5}, \frac{1}{5}\right) \text{ or } (0.2, 0.2)
\]
Let \( A(x_1, y_1) \) and \( B(x_2, y_2) \) be the coordinates of points \( A \) and \( B \). The coordinates of a point \( H \) that divides \( AB \) in the ratio \( m:n \) can be found using the formula:
\[
H\left(\frac{mx_2 + nx_1}{m+n}, \frac{my_2 + ny_1}{m+n}\right)
\]
In this case, \( m = 1 \) (for segment \( AH \)) and \( n = 4 \) (for segment \( HB \)). Therefore, the ratio \( AH:HB \) is \( 1:4 \).
This means \( H \) divides \( AB \) into 5 equal parts, with \( AH \) being 1 part and \( HB \) being 4 parts.
Assuming the coordinates of \( A \) and \( B \) are as follows (since specific coordinates are not given):
- Let \( A = (x_1, y_1) \)
- Let \( B = (x_2, y_2) \)
Using the section formula:
\[
H\left(\frac{1 \cdot x_2 + 4 \cdot x_1}{1+4}, \frac{1 \cdot y_2 + 4 \cdot y_1}{1+4}\right)
\]
This simplifies to:
\[
H\left(\frac{x_2 + 4x_1}{5}, \frac{y_2 + 4y_1}{5}\right)
\]
Now, we compare \( H \) with the provided coordinate options to find which one corresponds to this calculation.
However, without specific coordinates for \( A \) and \( B \), we must select an appropriate choice from the provided options based purely on ratio consideration and the requirement that the point divides \( AB \) in a 1:4 ratio.
Given the choices:
1. \( (-1, -2) \)
2. \( \left(\frac{1}{5}, \frac{1}{5}\right) \)
3. \( \left(\frac{3}{4}, \frac{3}{2}\right) \)
4. \( (0.6, 1.2) \)
If we calculate for \( H \) to make it in \( 1:4 \) ratio from \( A \) to \( B \), the fractions suggest that \( H \) should be closer towards point \( A \) if \( A \) is lower than \( B \) on the coordinate scale; since \( (0.2, 0.2) \) keeps both values low, it’s a strong candidate.
Therefore, the most logical option given these constraints would be \( \left(\frac{1}{5}, \frac{1}{5}\right) \) which reflects the lower ratio and maintains the proportional sequence \( AH \) and \( HB \).
### Final Answer:
\[
\left(\frac{1}{5}, \frac{1}{5}\right) \text{ or } (0.2, 0.2)
\]