Midpoint Theorem

In the figure, AB is parallel to PRQ, T and P are the midpoints of CQ and CA respectively and ART is a straight line. Prove that TR : RA = 1  : 2.

Solution

Point Q is the midpoint of BC (midpoint theorem)

QB : QC                 QC : TQ

   1 :  1                      2  :  1

      x2                          x1

   2  :  2                     2  :  1

QB  :  TQ

  2   :   1

∠TRQ = ∠TAB (corresponding RQ//AB)

TQ/QB = TR/RA

1/2 = TR/RA

Therefore TR : RA = 1 : 2

∠TQR = ∠TBA (corresponding ∠s, RQ//AB)

Therefore triangle TRQ is similar to triangle TAB

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