Question
Determine the coordinates of point J on AB¯¯¯¯¯¯¯¯ so that the ratio of AJ to JB is 1:1.(1 point)ResponsesJ(3,0)upper J left parenthesis 3 comma 0 right parenthesisJ(0,−3)upper J left parenthesis 0 comma negative 3 right parenthesisJ(0,3)upper J left parenthesis 0 comma 3 right parenthesisJ(0,2)
Answers
GPT-4o mini
To find the coordinates of point J on the line segment \( \overline{AB} \) such that the ratio of \( AJ \) to \( JB \) is \( 1:1 \), we need to determine the midpoint of the segment \( \overline{AB} \).
The midpoint \( J \) can be found using the formula:
\[
J\left(\frac{x_A + x_B}{2}, \frac{y_A + y_B}{2}\right)
\]
However, we need the coordinates of points \( A \) and \( B \) to proceed further. The provided options for the point \( J \) are:
- \( J(3, 0) \)
- \( J(0, -3) \)
- \( J(0, 3) \)
- \( J(0, 2) \)
If we assume that \( A \) and \( B \) are such that the midpoint \( J \) lies at any of these coordinate options, we can pick the correct answer based on the context of points \( A \) and \( B \).
Given that the question requires a 1:1 ratio and implies that \( J \) is equidistant from both \( A \) and \( B \), the correct answer depends on the specific coordinates of \( A \) and \( B \).
If \( A \) and \( B \) were at positions that allow \( J \) to take one of those coordinates, we can simply confirm that \( J(0, 2) \) does not imply a ratio since it does not equidistantly divide the segment unless further context is provided.
If we assume general case (since no specific coordinates for \( A \) or \( B \) were given), if J is at (3, 0) could be a possibility based on symmetry, but further information is needed about location.
**Therefore, without loss of context, confirm pair coordinates for A and B or assume:
**Final insight:** If A, B were symmetric around any of given coordinates, (3, 0) aligns better amidst given options as it appears balanced.
So, I select:
**J(3,0)** as the most plausible to be angle-out from close origin point denoting mid-measure.
The midpoint \( J \) can be found using the formula:
\[
J\left(\frac{x_A + x_B}{2}, \frac{y_A + y_B}{2}\right)
\]
However, we need the coordinates of points \( A \) and \( B \) to proceed further. The provided options for the point \( J \) are:
- \( J(3, 0) \)
- \( J(0, -3) \)
- \( J(0, 3) \)
- \( J(0, 2) \)
If we assume that \( A \) and \( B \) are such that the midpoint \( J \) lies at any of these coordinate options, we can pick the correct answer based on the context of points \( A \) and \( B \).
Given that the question requires a 1:1 ratio and implies that \( J \) is equidistant from both \( A \) and \( B \), the correct answer depends on the specific coordinates of \( A \) and \( B \).
If \( A \) and \( B \) were at positions that allow \( J \) to take one of those coordinates, we can simply confirm that \( J(0, 2) \) does not imply a ratio since it does not equidistantly divide the segment unless further context is provided.
If we assume general case (since no specific coordinates for \( A \) or \( B \) were given), if J is at (3, 0) could be a possibility based on symmetry, but further information is needed about location.
**Therefore, without loss of context, confirm pair coordinates for A and B or assume:
**Final insight:** If A, B were symmetric around any of given coordinates, (3, 0) aligns better amidst given options as it appears balanced.
So, I select:
**J(3,0)** as the most plausible to be angle-out from close origin point denoting mid-measure.