Answer:
(3, 6)
Explanation:
You want the coordinates of the orthocenter of triangle WXY with vertices W(2,7), X(3, 4), and Y(6, 7).
Orthocenter
The orthocenter is the point where the altitudes of the triangle meet. An altitude is the line through a vertex that is perpendicular to the opposite side.
This is an acute triangle, so the orthocenter is within the boundaries of the triangle.
Altitudes
We note that side WY of the triangle is a horizontal line, so the orthocenter will lie on the vertical line through point X. The x-coordinate of point X is 3, so that vertical line has equation x=3.
Side XY can be seen on the graph to have a slope of 1. Or, you can compute it from the slope formula:
m = (y2 -y1)/(x2 -x1)
m = (7 -4)/(6 -3) = 3/3 = 1
Then the point-slope equation of the altitude to that line will have the opposite reciprocal slope, and will go through point W(2, 7):
y -7 = -1(x -2)
Concurrent point
The y-value of the point of intersection of the altitude through W with the altitude through X will be the solution to ...
Substituting for x in the second equation, we have ...
y -7 = -3 +2
y = 6 . . . . . . . . . . add 7
The orthocenter is (x, y) = (3, 6).
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Additional comment
The orthocenter of a right triangle is the vertex where the right angle is located. The orthocenter of an obtuse triangle is outside the triangle.