Question

The steps shown can be used to prove that the medians of a triangle meet at a point.

Triangle A B C is shown on an x and y-axis. Lines are drawn from each point to the opposite side and intersect at point G. Point A is at (0, 0), point D is at (c, 0), point C is at (2 c, 0), point B is at (2 a, 2 b), and point E is at (a, b).

1. Define segments BD and CE as medians of triangle ABC.
2. Write linear equations for Line B D and Line C E.
3. Use a system of linear equations to solve for the coordinates of intersection point G.
4. Write the equation of Line A G.
5. Write an expression for the midpoint of BC, point F.
6. Show that point F lies on .

7. ?

Which is the final step to complete the proof?

Write a linear equation for each side of the triangle.
Write an expression for the midpoint of AC and BC.
Show that AF is congruent to BD and CE.
Show that AF is the median of BC.

Answer 1

Show that AF is the median of BC.

Reason: you have shown that G = BD ∩ CE and that the line AG passes through the midpoint F of BC. Therefore AG = AF is a median, so BD, CE and AF (the three medians) all meet at the same point G.