# What is the orthocenter of a triangle with corners at #(2 ,3 )#, #(5 ,1 )#, and (9 ,6 )#?

The Orthocenter is

The point where these two lines intersect is the orthocenter:

We solve for the x coordinate by setting the right sides equal since y = y.

Divide by two:

Divide by 5.

#x = 121/23

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To find the orthocenter of a triangle, you need to determine the point where the three altitudes of the triangle intersect. An altitude is a line segment from a vertex of the triangle perpendicular to the opposite side.

Given the coordinates of the triangle's vertices: ( A(2, 3) ), ( B(5, 1) ), and ( C(9, 6) ),

First, find the slopes of the lines connecting each pair of vertices to determine the slopes of the altitudes.

Then, find the equations of the perpendicular bisectors for each side of the triangle.

Finally, solve the system of equations formed by the perpendicular bisectors to find the orthocenter.

Alternatively, you can use the fact that the orthocenter is the intersection of the altitudes of the triangle, where an altitude is perpendicular to the opposite side. You can find the equations of the lines containing the altitudes and then solve for their point of intersection, which is the orthocenter.

The detailed calculations involve finding slopes, equations of lines, and solving equations. Would you like to proceed with the detailed calculations?

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