Q. 144.0( 338 Votes )

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 Δ ABC ~ Δ PQR

Answer :

To Prove: Δ ABC ∼ Δ PQR
Given: 

Proof: 


Let us extend AD and PM up to point E and L respectively, such that AD = DE and PM = ML.
Then, join B to E, C to

E, Q to L, and R to L

We know that medians divide opposite sides.

Hence, BD = DC and QM = MR

Also, AD = DE (By construction)

And, PM = ML (By construction)

In quadrilateral ABEC,

Diagonals AE and BC bisect each other at point D.

Therefore,

Quadrilateral ABEC is a parallelogram.

AC = BE and AB = EC (Opposite sides of a parallelogram are equal)

Similarly, we can prove that quadrilateral PQLR is a parallelogram and PR = QL, PQ = LR

It was given in the question that,



ΔABE ΔPQL (By SSS similarity criterion)

We know that corresponding angles of similar triangles are equal.

BAE = QPL   ..... (i)

Similarly, it can be proved that

ΔAEC ΔPLR and

CAE = RPL  ..... (ii)

Adding equation (i) and (ii), we obtain

BAE + CAE = QPL + RPL

⇒∠CAB = RPQ  .... (iii)

In ΔABC and ΔPQR,

(Given)

CAB = RPQ [Using equation (iii)]

ΔABC ΔPQR (By SAS similarity criterion)
Hence, Proved.

Rate this question :

How useful is this solution?
We strive to provide quality solutions. Please rate us to serve you better.
Related Videos
Basic Proportionality Theorem42 mins
Champ Quiz | Thales Theorem49 mins
NCERT | Strong Your Basics of Triangles39 mins
RD Sharma | Imp. Qs From Triangles41 mins
R.D Sharma | Solve Exercise -4.2 and 4.3FREE Class
R.D Sharma | Solve Exercise-4.545 mins
NCERT | Basic Proportionality Theorem22 mins
RD Sharma | Imp Qs Discussion- Triangles43 mins
5 Months Strategy For Maths Boards 202129 mins
Quiz | Criterion of Similarity of Triangle45 mins
Try our Mini CourseMaster Important Topics in 7 DaysLearn from IITians, NITians, Doctors & Academic Experts
Dedicated counsellor for each student
24X7 Doubt Resolution
Daily Report Card
Detailed Performance Evaluation
view all courses