Question

if point g is at (2,7) and point h is at (9,3) and point g and h is the endpoints of the hypotenuse of a right triangle what is the other point

Answer 1

what's the distance between g and h

Answer 2

To find the coordinates of the third point \( (x, y) \) that, with points \( G(2, 7) \) and \( H(9, 3) \), forms a right triangle where \( G \) and \( H \) are the endpoints of the hypotenuse, we can use the property that the lengths of the two legs are perpendicular to each other.

The slope of the line segment from \( G \) to \( H \) can be calculated as follows:

\[
\text{slope} = \frac{y_2 - y_1}{x_2 - x_1} = \frac{3 - 7}{9 - 2} = \frac{-4}{7}
\]

The slope of a line perpendicular to this would be the negative reciprocal:

\[
\text{slope} = \frac{7}{4}
\]

Let \( (x, y) \) be the coordinates of the third point, which is perpendicular to the segment \( GH \) at the midpoint \( M \) of \( GH \).

First, we find the midpoint \( M \):

\[
M = \left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \right) = \left( \frac{2 + 9}{2}, \frac{7 + 3}{2} \right) = \left( \frac{11}{2}, 5 \right) = (5.5, 5)
\]

Now we can go through two possible positions for point \( (x, y) \) based on the slope \( \frac{7}{4} \). The equations of the lines through the midpoint \( M(5.5, 5) \) will give us two points that satisfy being perpendicular.

The line equation in point-slope form is:

\[
y - y_1 = m(x - x_1)
\]

Substituting for one of the two options (for completeness let's say we are calculating one leg):

1. Using \( m = \frac{7}{4} \):

\[
y - 5 = \frac{7}{4}(x - 5.5)
\]

From here, you can choose a specific value of \( x \) to find \( y \) or vice versa.

Alternatively, using integer points and the knowledge of right triangle properties, the two orthogonal points can easily found through trials so they both meet at Pythagorean theorem setups. A rustic approach could derive points by simply meeting \( |x_a - x_b| \) or \( |y_a - y_b|\).

Doing this will yield possible integer coordinates for \( (x, y) \):

The possible coordinates of third points can fall near:
1. \( (9, 7) \) - vertically above \( H(9,3) \)
2. \( (2, 3) \) - horizontally level with \( H(9,3)\)
3. \( (5.5, 0) \) or provides for distinct right angle placements too.

Any of these can rigorously challenge upon plotting \( G, H \) and checking a right angled form. You may choose coordinates \( (7, 5) \) as per integer attempt if we are looking closely.

Thus, the three options (considering integers) may often circle closely as

**Possible 3rd Points:** \( (2, 3) \) and \( (9, 7) \) or even direct coordinates shifting can give refined options or approach!