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The sides AB and BC and the median AD of...

The sides AB and BC and the median AD of triangle ABC are equal to the sides PQ and QR and the median PM of triangle PQR respectively. Prove that the triangles ABC and PQR are congruent.

Text Solution

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According to the given statement AD is a median of `DeltaABC`.
`:.` D is the mid-point of BC
`implies BD=DC=1/2 BC` ...(1)
Similarly, PM is the median of `DeltaPQR`.
`:.` M is the mid - point of QR
`implies QM=MR=(QR)/2` ...(2)
It is given that `BC=QR`
`:. BD=QM` [from eqs. (1) and (2)]
In `Deltas ABD` and `PQM`,
`:' {(AB=PQ,"(given)"),(AD=PM,"(given)"),(BD=QM,"(just proved)"):}`
`:. DeltaABD cong DeltaPQM` (SSS)
`implies" "angleB=angleQ` (c.p.c.t)
Now, in `Deltas ABC` and PQR
`:' {(AB=PQ,"(given)"),(angleB=angleQ,"(proved above)"),(BC=QR,"(given)"):}`
`:. DeltaABC cong DeltaPQR` (SAS) Hence Proved.
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