Lena W.

asked • 12/12/16

Find the orthocenter of a triangle if...

Find the orthocenter of triangle ABC if
A=(0,0)
B=(4,0)
C=(4,2)
 
I've tried this problem on my own several times, and being homeschooled I'm a bit short on help. The zeros are throwing me off!

Kenneth S.

You should be grateful for the zeros.
 
The centroid (G) is the point of intersection of the three medians. From any vertex, a line segment to the midpoint of the opposite side is a median.
The centroid divides each median into two segments such that the distance from the vertex to the centroid is ? of the entire length of the median. In the figure, if BG = 2, then GA = 1, as labeled. In general, BG = 2(GA).
 
WITH THIS INFORMATION, YOU CAN FIND EQUATIONS OF AM (M being midpoint of BC, so M's coordinates are (4,2))
AND BN (N being midpoint of AB, so N's coordinates are (2,0))
 
Then you find the point of interesection of those two lines.
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12/12/16

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Kenneth S. answered • 12/12/16

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For line AM (as defined in my Comment) I get y = ½x
 
For line CN I get y - 4 = 4(x -4 ) or y = 2x - 4.
 
Setting these two y expression equal, I got x = 8/2 and so y = 4/3.
 
So I've done some work for you.  Your job is to CHECK THE WORK--if you see anything amiss, let me know.

I think I did excellent work, BUT UNFORTUNATELY I SOLVED FOR CENTROID & you asked for ORTHOCENTER. I'll submit another COMMENT (containing proper guidance for the actual problem).

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Kenneth S.

An altitude is a line segment constructed from a vertex and intersecting the opposite side (or its extension) perpendicularly.

The orthocenter (H) is the point of intersection of the three altitudes of a triangle. That is, the altitudes always meet at one point, the orthocenter. You can remember this if you know that ‘orthogonal’ is a fancy word for perpendicular, so the three altitudes, each one.
 
So to solve this,you will find equations of the altitude passing through M and N (midpoints of legs of your right triangle)-this is easy since one altitude gives an x value, the other gives the y value.
 
You only need two altitudes (the third one will pass through the same orthocenter, of course).  
 
It looks as if the answer is Orthocenter: (2,2)...A lot easier than the Centroid problem!
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12/12/16

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