Question
if point g is at (2,7) and point h is at (9,3) and that is the hypotenuse of a right triangle what is the other point
Answers
GPT-4o mini
To find the coordinates of the third point, which forms a right triangle with points \( G(2,7) \) and \( H(9,3) \) as the endpoints of the hypotenuse, we need to determine the midpoint of segment \( GH \) and then find a point that is perpendicular to line \( GH \).
1. **Calculate the midpoint \( M \) of segment \( GH \)**:
\[
M = \left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \right) = \left( \frac{2 + 9}{2}, \frac{7 + 3}{2} \right) = \left( \frac{11}{2}, 5 \right) = \left( 5.5, 5 \right)
\]
2. **Find the slope of line segment \( GH \)**:
\[
\text{slope} = \frac{y_2 - y_1}{x_2 - x_1} = \frac{3 - 7}{9 - 2} = \frac{-4}{7}
\]
3. **Determine the slope of the perpendicular line**:
The slope of a line that is perpendicular to another line is the negative reciprocal. Thus, for a slope of \( -\frac{4}{7} \), the perpendicular slope is:
\[
\text{perpendicular slope} = \frac{7}{4}
\]
4. **Using the midpoint and the perpendicular slope, we can find a point that is a right angle**:
If we move a distance \( d \) from the midpoint \( (5.5, 5) \) along the direction of the perpendicular slope, we can find a point:
Let's say we choose \( d = 1 \) for simplicity, the change in \( x \) (run) and change in \( y \) (rise) based on the slope can be calculated as:
- Change in \( x \): \( 1 \cdot \frac{4}{\sqrt{4^2 + 7^2}} \)
- Change in \( y \): \( 1 \cdot \frac{7}{\sqrt{4^2 + 7^2}} \)
The hypotenuse would create two right angles, one above and one below the line connecting \( G \) to \( H \). Calculating these two alternative points, we can choose to use simple integer values or fractions based on the distance from the midpoint.
**Finding the new points**:
- One point \( P_1 \):
\[
P_1 = (5.5 + \frac{4}{\sqrt{65}}, 5 + \frac{7}{\sqrt{65}})
\]
- Another point \( P_2 \):
\[
P_2 = (5.5 - \frac{4}{\sqrt{65}}, 5 - \frac{7}{\sqrt{65}})
\]
Calculating these gives:
- The approximate values will give the new points depending on chosen simple integer heights while maintaining a perpendicular length around \( M \).
Thus, theoretically, there can be two solutions (two points perpendicular) that could serve as the vertex of the right triangle based on symmetry on either side of line \( GH \), and for exact values found, or visual representation can yield actual locations.
Instead for a direct site answering without much calculation, we would conventionally state that the location of point can be obtained by following perpendicular directions traced out in the graph program.
However, without additional context of specific limits, if we need to specify exact locations you may simply refine an arbitrary distance and find coordinates from the hypotenusal midpoint traveling a uniform distance calculated from vertical shifts on graphical setups will give adequate answer projections.
1. **Calculate the midpoint \( M \) of segment \( GH \)**:
\[
M = \left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \right) = \left( \frac{2 + 9}{2}, \frac{7 + 3}{2} \right) = \left( \frac{11}{2}, 5 \right) = \left( 5.5, 5 \right)
\]
2. **Find the slope of line segment \( GH \)**:
\[
\text{slope} = \frac{y_2 - y_1}{x_2 - x_1} = \frac{3 - 7}{9 - 2} = \frac{-4}{7}
\]
3. **Determine the slope of the perpendicular line**:
The slope of a line that is perpendicular to another line is the negative reciprocal. Thus, for a slope of \( -\frac{4}{7} \), the perpendicular slope is:
\[
\text{perpendicular slope} = \frac{7}{4}
\]
4. **Using the midpoint and the perpendicular slope, we can find a point that is a right angle**:
If we move a distance \( d \) from the midpoint \( (5.5, 5) \) along the direction of the perpendicular slope, we can find a point:
Let's say we choose \( d = 1 \) for simplicity, the change in \( x \) (run) and change in \( y \) (rise) based on the slope can be calculated as:
- Change in \( x \): \( 1 \cdot \frac{4}{\sqrt{4^2 + 7^2}} \)
- Change in \( y \): \( 1 \cdot \frac{7}{\sqrt{4^2 + 7^2}} \)
The hypotenuse would create two right angles, one above and one below the line connecting \( G \) to \( H \). Calculating these two alternative points, we can choose to use simple integer values or fractions based on the distance from the midpoint.
**Finding the new points**:
- One point \( P_1 \):
\[
P_1 = (5.5 + \frac{4}{\sqrt{65}}, 5 + \frac{7}{\sqrt{65}})
\]
- Another point \( P_2 \):
\[
P_2 = (5.5 - \frac{4}{\sqrt{65}}, 5 - \frac{7}{\sqrt{65}})
\]
Calculating these gives:
- The approximate values will give the new points depending on chosen simple integer heights while maintaining a perpendicular length around \( M \).
Thus, theoretically, there can be two solutions (two points perpendicular) that could serve as the vertex of the right triangle based on symmetry on either side of line \( GH \), and for exact values found, or visual representation can yield actual locations.
Instead for a direct site answering without much calculation, we would conventionally state that the location of point can be obtained by following perpendicular directions traced out in the graph program.
However, without additional context of specific limits, if we need to specify exact locations you may simply refine an arbitrary distance and find coordinates from the hypotenusal midpoint traveling a uniform distance calculated from vertical shifts on graphical setups will give adequate answer projections.