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/
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