Two sides AB and BC and median AM of one triangle ABC are respectively equal to sides PQ and QR and median PN of ΔPQR (see Fig. 7.40). Show that:
(i) Δ ABM ≅ Δ PQN
(ii) Δ ABC ≅ Δ PQR
Solution:
Given: AB = PQ, AM = PN, BM = QN
(i) In ΔABC, AM is the median to BC.
∴ BM = 1/2 BC
In ΔPQR, PN is the median to QR.
∴ QN = 1/2 QR
It is given that BC = QR
∴ 1/2 BC = 1/2 QR
∴ BM = QN … (1)
In ΔABM and ΔPQN,
AB = PQ (Given)
BM = QN [From equation (1)]
AM = PN (Given)
∴ ΔABM ≅ ΔPQN (Using SSS congruence criterion)
⇒ ∠ABM = ∠PQN (By CPCT)
⇒ ∠ABC = ∠PQR … (2)
(ii) In Δ ABC and Δ PQR ,
AB = PQ (Given)
∠ABC = ∠PQR [From Equation (2)]
BC = QR (Given)
∴ ΔABC ≅ ΔPQR (By SAS congruence rule)
☛ Check: NCERT Solutions Class 9 Maths Chapter 7
Video Solution:
Two sides AB and BC and median AM of one triangle ABC are respectively equal to sides PQ and QR and median PN of ΔPQR (see Fig. 7.40). Show that: (i) Δ ABM ≅ Δ PQN (ii) Δ ABC ≅ Δ PQR
NCERT Maths Solutions Class 9 Chapter 7 Exercise 7.3 Question 3
Summary:
If two sides AB and BC and median AM of one triangle ABC are respectively equal to sides PQ and QR and median PN of ΔPQR, then ΔABM ≅ ΔPQN by SSS congruence and ΔABC ≅ ΔPQR by SAS congruence.
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