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Median and Altitude Constructions

Date: 07/16/2003 at 05:04:44
From: Karam Bir Singh
Subject: Constructions

Dear Sir,

I know the altitudes, medians, orthocenter, in the circle, centroid, 
and circumcircle. But I do not know how to draw a median and an 
altitude from the three sides of the triangle. I do not know the 
steps of constructions, i.e. where the compass should be kept and all 
that, in order to draw the altitudes and the medians. 


Date: 07/16/2003 at 09:19:10
From: Doctor Jaffee
Subject: Re: Constructions

Hi Karam,

There are two ways to construct isosceles triangles that will help you 
construct a median and an altitude of any triangle.

First, start with any segment and place the point of the compass at an 
endpoint. Set the radius of the compass at more than half the length 
of the segment. Draw an arc. Then put the point of the compass at the 
other endpoint of the segment, don't change the radius, and draw 
another arc.  Where the two arcs meet will be the vertex of the 
isosceles triangle.  Connect that point to each of the endpoints of 
the original segment and you have your triangle. Now, construct 
another isosceles triangle using the same segment, but a different 
radius setting.

When you draw the line that goes through the two vertices you have 
constructed, the line will also be the perpendicular bisector of the 
original segment. That is why this construction is useful in finding 
medians and altitudes.

Now, suppose you want to construct the medians of the triangle ABC. 
Use the construction method I have described with AB being the 
original segment.  When you are finished, you will have the midpoint 
of AB. The segment that connects that point to C will be a median. 
Repeat this process with the other two sides.

Suppose you have a line XY and a point, P, not on the line. If you 
place the point of your compass on P and draw an arc that intersects 
the line XY in two points, Q and R, then PQR will be an isosceles 
triangle. Use QR as the base of the triangle and use the method I 
explained in the second paragraph to construct another isosceles 
triangle. When you draw a line through the vertex of this triangle and 
the point P, the line will be perpendicular to XY. In other words, you 
will have a line that passes through P and is perpendicular to XY.

So, let's get back to triangle ABC. If you want to construct the 
altitude from C, use the procedure that I explained in the previous 
paragraph.

Give it a try and if you want to check your constructions with me or 
if you have difficulties or other questions write back and I'll try to 
help you some more.

Good luck,

- Doctor Jaffee, The Math Forum
  http://mathforum.org/dr.math/ 
Associated Topics:
High School Triangles and Other Polygons

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