# Investigating the Centroid using Transformations

• using the polygon tool (5th tool box from left), draw triangle ABC
• using the midpoint tool (2nd tool box from left), plot the midpoint of segment AB, segment BC and segment CA
• using the segment tool (3rd tool box from the left), draw the medians of triangle ABC
• using the intersect tool (2nd tool box from left), create point G at the intersection of the medians
• locate the transformation tool box (4th from right)
• rotate triangle DEF 180 degrees about point G to create triangle D'E'F'
• dilate triangle D'E'F', center of dilation G, scale 2 to create D"E"F"
• in "options", select "labeling" and turn on "all new objects"
• draw segment FG, segment GF' and segment F'F" to create j, k and l
• using object properties, recolor each segment

1. Compare the lengths of segments j, k and l. What do you notice?
2. Using the move tool (left tool box), drag A. Does the relationship between the segments change?
3. Why does FG = GF'?
4. Why is GF" = 2(GF')?
5. What is the ratio between FG and GF"?
6. Does this ratio hold true for the segments of the other medians?
7. Rename point G "centroid".
8. Write a sentence containing something you discovered about the centroid in this activity.

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