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In ΔPQR seg PM is a median. Angle bisectors of ∠PMQ and ∠PMR intersect side PQ and side PR in points X and Y respectively. Prove that XY || QR. Complete the proof by filling in the boxes.

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Sol.       In ΔPMQ ray MX is a bisector of ∠PMQ.

∴           \({PM\over MQ} = {PX\over XQ}\)           ...(I) [Theorem of angle bisector]

            In ΔPMR, ray MY is bisector of ∠PMR.

∴            \({PM\over MR }={ PY\over YR}\)          ...(II) [Theorem of angle bisector]

              QM = MR           ...(III) [M is the midpoint of seg QR]

             \({PM\over MR} ={ PX\over XQ}\)           ...(IV) [From (I) and (III)]

∴           \({PX\over XQ }={ PY\over YR}\)                     [From (I), (II) & (IV)]

∴          XY || QR                        [Converse of Basic Proportionality Theorem]

Similarity August 22 , 2018 0 Comments 16 views