Question

Question 12 Sides AB and BC and median AD of a triangle ABC are respectively proportional to sides PQ and QR and median PM of ΔPQR (see the given figure). Show that ΔABC ∼ ΔPQR.

Class 10 - Math - Triangles Page 141

Answer 1

Given: ΔABC and ΔPQR, AB, BC and median AD of ΔABC are proportional to sides PQ, QR and median PM of ΔPQR

i.e., AB/PQ = BC/QR = AD/PM


To Prove: ΔABC ~ ΔPQR


Proof: AB/PQ = BC/QR = AD/PM



⇒ AB/PQ = BC/QR = AD/PM (D is the mid-point of BC. M is the mid point of QR)

⇒ ΔABD ~ ΔPQM [SSS similarity criterion]

∴ ∠ABD = ∠PQM [Corresponding angles of two similar triangles are equal]

⇒ ∠ABC = ∠PQR

In ΔABC and ΔPQR

AB/PQ = BC/QR ...(i)

∠ABC = ∠PQR ...(ii)

From equation (i) and (ii), we get

ΔABC ~ ΔPQR [By SAS similarity criterion]

Answer 2

Answer:

Step-by-step explanation: please look at the attachment for better understanding.

Hence ΔABD ~ ΔPQM (SSS similarity criterion)

∴ ∠B = ∠Q

Since, D and M are the mid points of side BC and QR resp.

BC = QR ( 2BD=2QM)

And AB = PQ (given)

∴ ΔABC ~ ΔPQR (SAS similarity criterion)