# geometry

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I need to find the orthocenter of the triangle with the vertices R(-4,8), S(-1,5), T(5,5)???
How do you get 'y' for this problem?

• geometry -

One way to solve this problem is to think of the altitudes from R,S,T to the opposite side. For each of these lines, we have a point and a slope (perpendicular to the line forming the opposite side)

So, a nice easy place to start is to notice that ST is a horizontal line. So, the altitude from R is the line

x = -4

Now, if we find where another altitude intersects that line, we are done.

The slope of RT is (5-8)/(5+4) = -3/9 = -1/3

The altitude from S to RT will have slope 3.

So, now we have a point (-1,5) and a slope (3).

(y-5)/(x+1) = 3
y = 3x + 8

That intersects x=-4 at y=4

So, the orthocenter is at (-4,4)

If you want, you can verify by checking the other altitudes.

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