Sides AB and AC and median AD of a triangle ABC are respectively proportional to side PQ and PR and median PM of another triangle PQR. Show that triangle ABC is similar to triangle PQR.
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Sides AB and AC and median AD of a triangle ABC are respectively proportional to side PQ and PR and median PM of another triangle PQR. Show that triangle ABC is similar to triangle PQR.
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(Image 1)
Given: △ABC & △PQR
AD is the median of △ABC
PM is the median of △PQR
PQAB=PRAC=PMAD→1
To prove: △ABC∼△PQR
Proof:Let us extend AD to point D such that that AD=DE and PM upto point L such that PM=ML
Join B to E, C to E, & Q to L and R to L
(Image 2)
We know that medians is the bisector of opposite side
Hence BD=DC & AD=DE *By construction)
Hence in quadrilateral ABEC, diagonals AE and BC bisect each other at point D
∴ABEC is a parallelogram
∴AC=BE & AB=EC (opposite sides of a parallelogram are equal) →2
Similarly we can prove that
PQLR is a parallelogram.
PR=QL,PQ=LR (opposite sides of a parallelogram are equal) →3
Given that
PQAB=PRAC=PMAD (frim 1)
⇒PQAB=QLBE=PMAD (from 2 and 3)
⇒PQAB=QLBE=2PM2A