Three medians, one point
Median — a line from a corner to the far side's midpoint. A triangle has three medians, and they meet at one point.
- Triangle ABC, with all three medians drawn.
- The medians meet at one point, G.
- G is called the centroid.
- G splits each median 2 to 1, counted from the corner.
The rule
Rule: The three medians meet at one point, the centroid. It splits every median 2 to 1 from the corner.
Why? Take medians BE and CF, which cross at G. E is the midpoint of AC. F is the midpoint of AB. Let P and Q be the midpoints of GB and GC.
FE joins two midpoints of triangle ABC. So FE is half of BC, and parallel to it. PQ joins two midpoints of triangle GBC. So PQ is half of BC, and parallel to it. FE and PQ are equal and parallel. So FEQP is a parallelogram.
The diagonals of FEQP are FQ and EP. They lie along the medians CF and BE. So they cross at G. They bisect each other: each cuts the other in half. So EG equals GP. And BP equals PG, since P is the midpoint of GB. That gives three equal parts on BE: BP, PG and GE. So BG is 2 parts and GE is 1 part.
Now take the medians AD and BE. D is the midpoint of BC. The same steps work for them. The median AD also cuts BE 2 to 1 from B. Only one point of BE does that. It is G. So all three medians meet at G.
| From the corner | To the midpoint |
|---|---|
| 2 parts | 1 part |
With coordinates, add the three x-values and divide by 3. Do the same for y. Take corners (0, 0), (6, 0) and (3, 9). The x-values total 9, and 9 ÷ 3 = 3. The y-values total 9, and 9 ÷ 3 = 3. The centroid is (3, 3).
Read each question. Pick the length or the point.
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