Question

In the figure, two sides AB, BC and the median AD of ΔABC are respectively equal to two sides PQ, QR and median PS of ΔPQR. Prove that
(i) ΔADB ≅ ΔPSQ;
(ii) ΔADC ≅ ΔPSR.
Does it follow that triangles ABC and PQR are congruent?

Answer 1

(i) In ΔABC, AM is the median to BC.

∴ BM =BC

In ΔPQR, PN is the median to QR.

∴ QN =QR

However, BC = QR

∴BC =QR

⇒ BM = QN (1)

In ΔABM and ΔPQN,

AB = PQ (Given)

BM = QN [From equation (1)]

AM = PN (Given)

∴ ΔABM ≅ ΔPQN (SSS congruence rule)

ABM = PQN (By CPCT)

ABC = PQR (2)

(ii) In ΔABC and ΔPQR,

AB = PQ (Given)

ABC = PQR [From equation (2)]

BC = QR (Given)

⇒ ΔABC ≅ ΔPQR (By SAS congruence rule)

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Answer 2

Given :

Two sides AB, BC and the median AD of ΔABC are respectively equal to two sides PQ, QR and median PS of ΔPQR.

To prove :

(i) ΔADB ≅ ΔPSQ;
(ii) ΔADC ≅ ΔPSR

Proof :

i ) In ∆ADB and ∆PSQ

AB = PQ

AD = PS

BD = QS

[ BC = QR => BC/2 = QR/2

=> BD = QS and DC = SR ]

Therefore ,

ΔADB ≅ ΔPSQ

[ SSS congruence rule ]

ii ) In ∆ADC and ∆PSR

AD = PS ( side )

<ADC = <PSR [ Angle ]

DC = SR ( side )

Therefore ,

ΔADC ≅ ΔPSR

[ SAS congruence rule ]

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