# An Investigation ofMedians in a Triangle

## Background

In this section of the exploration you will be asked to discover the way the medians of a triangle relate to each other. As you work through the material, you may be asked to write down the answers to the questions in the investigation or to save the information and your sketch on a disk. Ask your teacher for specific directions.

## Exploring the Medians of a Triangle

Click on the image below to activate the Geometer's Sketchpad Triangle Sketch. You are going to use this sketch to construct the medians of a triangle. The instructions for the construction are below the picture. Following the construction, you will use your sketch to investigate relationships involving medians.

## Construct

• Construct the median to side AC using a script. Click here to access it. Then select the points A,B and C in the given order and play the script.
• Construct a second median to side AB following the procedure below.
• Construct the midpoint of segment AB.
• Draw a segment from vertex C (the vertex opposite that segment) to midpoint G.
• Make a point at the intersection of these two medians.
• Construct the third median to side BC in the same manner.

## Investigate

1. When you constructed the third median, did it pass through the same point as the other two medians?
2. Use the selection tool (the arrow) to move the vertices of the original triangle. Does the way the medians intersect change?
3. Write a conjecture about the way the medians intersect in a triangle.

We call the point of intersection of the medians in a triangle the centroid.

## Construct

Click on the image below to activate another Geometer's Sketchpad Triangle Sketch. You are going to use this sketch to measure the lengths of the medians and the segments created by their intersection.
• Measure the length of segment AI.
• Construct segment AH and measure its length.
• Construct segment HI and measure its length.
• Measure the lengths of the other medians and their parts in the same manner.

## Investigate

1. Calculate the ratios AH/AI, CH/CG, and BH/BF. What do you notice about the ratios? Does this change when you move the triangle?
2. Calculate the ratios AH/HI, BH/HF, and CH/HG. Make a conjecture about the way the centroid H divides the medians of a triangle.