Question 1062408: P is the midpoint of segment BC in triangle ABC and Q is the midpoint of seg AP .ray BQ cuts segment AC at R. show that : BQ = 3QR
Answer by KMST(5328)   (Show Source):
You can put this solution on YOUR website! Here is a triangle ABC, including points P, Q, R, a few extra midpoints, and a midsegment.
 S, T, and U are the midpoints of AC, AB, and PC respectively.
Of course, ST is the midsegment of ABC, and SQ is the midsegment of APC.
Since midsegments are half a long as the base,  .
Since SQ and UC are congruent and parallel,
SQUC is a parallelogram,
and QU is parallel to AC .
With their pairs of parallel sides, triangles SQR and UBQ have 3 pairs of congruent angles.
That makes triangles SQR and UBQ similar triangles.
Since  ,  .
The corresponding side in SQR is  ,
So, sides of UBQ are  times longer than corresponding sides of SQR :
UB = 3SQ , UQ = 3SR , and  .
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