PROVE THE FOLLOWING RESULT: Using a compass and a straightedge, construct a triangle ABC such that...

PROVE THE FOLLOWING RESULT: Using a compass and a straightedge, construct a triangle ABC such that side BC is of length 2, the circumradius is of length 3/2 and the median AA' is of length 2 and is the midpoint of BC.

Solutions

Expert Solution

We can use the definitions of median and circumradius to construct the given triangle.

First we draw the side BC and mark its midpoint A'. As the median from A to A' has length 2, all points lying at distance 2 from A' are suitable candidates for the third vertex. These are just the points on the circle of radius 2 with center A'.

Then, we use the circumradius. If we draw a circle with radius 3/2 and let it be the circumcircle for our triangle, BC is a chord of length 2 for this circle. In addition, the point A also has to lie on this circle.

So, the third vertex has to be the points lying on both these circles, that is, the point of intersection of the 2 circles.

After drawing the triangle, we can easily measure and verify that AA' has the required length.

Colby Messinger answered 1 year ago

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