# Triangle congruent problem

1. Apr 29, 2007

### Styx

In the diagram, AM is a medium of triangle ABC. Perpendicular lines drawn from B and C to AM (or its extension) meet AM at P and Q respectively.

Prove that BP = CQ

So far I have concluded that:

BM = CM
Angle BPM = angle CQM
Triangle ABM = ACM

I am not sure what else I can do in order to prove that the triangles BMP and CMQ are congruent which would prove that BP = CQ

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Last edited: Apr 29, 2007
2. Apr 30, 2007

### HallsofIvy

Staff Emeritus
Since the lines are drawn perpendicular to AM, you have RIGHT triangles. Further, since M is a "median" (note spelling) BM and CM are congruent.

3. Apr 30, 2007

### Styx

Consider Triangles BMP and CMQ
You know

BM = CM (given hypothesis)

angle BPM = angle CQM as they are both at right angles to AM or its extension

180 degrees - angle AMC = angle AMB, angle AMB + angle QMB = 180 degrees
Therefore, angle AMC = angle AMB

Triangle BMP and CMQ are congruent by the AAS condition of the triangle theorm.

Therefore, BP = CQ

Last edited: Apr 30, 2007