Medians! Medians! Everywhere!

Assignment 6

By Amber Candela

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The Problem

Construct a triangle and its medians. Construct a second triangle with the three sides having the lengths of the three medians from your first triangle. Find some relationship between the two triangles.

What is a median?

A median is the segment that connects the vertex to the midpoint of the opposite side. The medians intersect to create segments that are in a ratio of 2:1. The median bisects the triangle, creating two smaller triangles that have the same area.

How can you construct this?

First create triangle ABC with medians AD, FC and BE. We need to create a triangle with the medians, so start the new triangle with median FC. We then need to use circles to construct the other two sides of the new triangle. Construct circle with center F and the radius the same length as median AD. Then construct circle with center C and the radius the same length as median BE. The two circles intersect at point G. Create triangle FGC which will be constructed with the median lengths from triangle ABC. Side FG = AD and side GC = BE by construction and side FC = FC by the reflexive property. The construction can be seen below.

What are the relationships between triangle ABC and triangle FGC?

Are the triangles congruent?

No. The sides of the original triangle are not equal to the sides of the new triangle, thus by SSS the triangles are not congruent. This would mean they do not have the same perimeter. The areas of the triangles are also not equal because they do not have the same base or height.

The ratio areas of the two triangles is constant. The areas are in a ratio of 4:3. Use the GSP file below to explore the area ratios.

Area Ratio Explorer

Proof of the ratios.

While GSP is useful to show how the areas are in the ratio of 4:3, it cannot prove such statements. Let's look at a the triangles in a different way.

Notice this time we have two pairs of parallel lines, GF is parallel to EB and GC is parallel to AD. This created two parallelograms, FBEG and AGCD.

How do these parallelograms help?