Ex 6.3, 14
Sides AB and AC and median AD of a triangle ABC are respectively proportional to sides PQ and PR and median PM of another triangle PQR. Show that ABC PQR.
Given: ABC and PQR
AD is the median of ABC
,PM is the median of PQR
/ = / = /
To Prove:- ABC PQR.
Proof:
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 is the bisector of opposite side
Hence, BD = DC
Also, AD = DE
Hence in quadrilateral ABEC,
diagonals AE and BC bisect each other at point D.
Therefore, quadrilateral ABEC is a parallelogram.
AC = BE and AB = EC
Similarly, we can prove that
PQLR is a parallelogram
PR = QL, PQ = LR
Given that
/ = / = /
/ = / = /
/ = / =2 /2
/ = / = /
ABE PQL
ABE PQL
We know that corresponding angles of similar triangles are equal.
BAE = QPL
Similarly, we can prove that
AEC PLR
We know that corresponding angles of similar triangles are equal.
CAE = RPL
Adding (4) & (5),
BAE + CAE = QPL + RPL
CAB = RPQ
In ABC and PQR,
/ = /
CAB = RPQ
ABC PQR
Hence proved

Davneet Singh has done his B.Tech from Indian Institute of Technology, Kanpur. He has been teaching from the past 13 years. He provides courses for Maths, Science, Social Science, Physics, Chemistry, Computer Science at Teachoo.

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