# What is the orthocenter of a triangle with corners at #(1 ,3 )#, #(5 ,7 )#, and (9 ,8 )#?

We only need the equations of two lines to find the orthocenter of a triangle, which is the point where the line of the heights relative to each side (passing through the opposed vertex) meets.

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To find the orthocenter of a triangle, you need to find the intersection point of the altitudes of the triangle. An altitude is a line perpendicular to a side of the triangle and passing through the opposite vertex. To find the altitude passing through a vertex, you need to find the equation of the line perpendicular to the side opposite that vertex and passing through the vertex.

First, find the slopes of the sides of the triangle using the given coordinates of the vertices. Then, use these slopes to find the slopes of the perpendicular lines (altitudes). After finding the equations of the altitudes, solve the system of equations to find their intersection point, which is the orthocenter of the triangle.

The equation of the altitude passing through a vertex can be found using the point-slope form of a line: (y - y_1 = m(x - x_1)), where (m) is the slope of the perpendicular line and ((x_1, y_1)) is the vertex through which the altitude passes.

Once you have the equations of two altitudes, solve them simultaneously to find the orthocenter.

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