Question

In two triangles, if
two sides and medians to the
third sides are
equal then two triangles
aru
congruent .​

Answer 1

Answer:

Given: △ABC and △PQR where BD and QS are the medians and AB = BC = BD

PQ = QR = QS

To prove: △ABC ~ △PQR

Construction: Produce BD and QS to E and T respectively such that BD = DE

and QS = ST. CE and TR are joined.

Proof: In △ADB and △CDE,

AD = DC (given)

△ADB = △CDE (Vertically opposite angles)

BD = DE.

△ADB ≅ △CDE (SAS ≅ axiom)

Hence AB = CE and ∠ABD = ∠DEC.

Similarly △PQS ≅ △RST,

hence PQ = TR and ∠PQS = ∠STR.

Consider △EBC and △TQR,

BD = 2BD = BE (from given and construction)......(1)

BD/QS = 2BD/2QS = BE/QT (from given and construction).......(1)

AB = CE and PQ = RT(proved),

AB/PQ = CE/RT....(2)

AB/PQ = BC/QR = BD/QS ....(3)

From (1),(2)and(3),

BE/QT = CE/RT = BC/QR

∴△EBC ~ △TQR (SSS similarity axiom).

⇒ ∠DBC = ∠SQR and ∠DEC = ∠STR ....(4)(corresponding angles of similar triangles

are proportional)

But∠ABD = ∠DEC and ∠PQS = ∠STR(Proved)....(5)

∴ ∠ABD = ∠PQS (from(4)and(5))....(6)

From (5) and (6),

∠ABC = ∠PQR ....(1)

In △ABC and △PQR,

AB/PQ = BC/QR (given)

And ∠ABC = ∠PQR (from 1)

∴ △ABC ~ △PQR (SAS Similarity).