two sides and median of one triangle are respectively equal to two sides and median of other triangle. prove that triangle are congruent
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Answers
Answer:
Let us extend AD to point D such that AD=DE and PM up to point L such that PM=ML
Join B to E,C to E,Q to L and R to L
We know that medians are the bisector of the opposite side
Hence BD=DC
Also,AD=DE (by construction)Hence in quadrilateral ABEC, diagonals AE and BC bisect each other at point D
∴ quadrilateral ABEC is a parallelogram.
∴AC=BE and AB=EC .......(1) since opposite sides of a parallelogram are equal.
We know that medians are the bisector of the opposite side
Hence QM=MR
Also, PM=ML (by construction)Hence in quadrilateral PQLR, diagonals PL and QR bisect each other at point M
∴ quadrilateral PQLR is a parallelogram.
∴PR=RL and PQ=QL ........(1)since opposite sides of a parallelogram are equal.
Given:
PQ
AB
=
PR
AC
=
PM
AD
⇒
PQ
AB
=
QL
BE
=
PM
AD
from (1) and (2)
⇒
PQ
AB
=
QL
BE
=
2PM
2AD
⇒
PQ
AB
=
QL
BE
=
PL
AE
as AD=DE
AE=AD+DE=AD+AD=2AD
PM=ML
PL=PM+ML=PM+PM=2PM
∴△ABE∼△PQL by SSS similarity criterion
∴△ABE∼△PQL and △AEC∼△PLR
We know that corresponding angles of similar triangles are equal
∴∠BAE=∠QPL ........(3)
and ∠CAE=∠RPL ........(4)
Adding (3) and (4) we get
∠BAE+∠CAE=∠QPL+∠RPL
∠CAB=∠RPQ ........(5)
In △ABC and △PQR
PQ
AB
=
PR
AC
∠CAB=∠RPQ
∴△ABC∼△PQR by SAS similarity criterion
Hence proved.
Step-by-step explanation:
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Answer:
2 yes
Step-by-step explanation:
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