SOLUTION: Given a parallelogram ABCD, side AB is extended to a point P such that AB : BP = 2 : 1 and side CD is extended to a point Q such that D is the midpoint of CQ. The line PQ meets t

Question 1164923: Given a parallelogram ABCD, side AB is extended to a point P such that
AB : BP = 2 : 1 and side CD is extended to a point Q such that D is the
midpoint of CQ. The line PQ meets the diagonal AC at Z and the sides
BC and AD at X and Y , respectively.
(a) Find the ratio of the areas of triangles AZY and CZX.
(b) Find the ratio of the lengths PX and ZY.
I am confused with this. Do you know how to do it
Answer by solver91311(24713) (Show Source): You can put this solution on YOUR website!

I'll give you a hint. Angle AZP is congruent to Angle CZQ by vertical angles. Angle DAC is congruent to angle BCA by opposite interior angles of a transversal and angles BAD and BCD are congruent for the same reason, but since angle BAD is the sum of angle BAC and CAD, and angle BCD is the sum of BCA and ACD, angles ACD and BAC are congruent. And APZ is congruent to CQZ by opposite interior angles. Therefore triangles APZ and CQZ are similar by AAA. Using a similar argument, triangles AZY and CZX are similar by AAA. Then AP:CQ is 3:4. You can take it from there.

John

My calculator said it, I believe it, that settles it

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