Q8. Two sides AB and BC and median AM of one triangle ABC are
respectively equal to sides PQ and QR and median PN of ∆ PQR. Show
that
i) ∆ ABM ∆ PQN.
ii) ∆ ABC ∆ PQR
Answers
Answer:
Step-by-step explanation:
Congruence of triangles:
Two ∆’s are congruent if sides and angles of a triangle are equal to the corresponding sides and angles of the other ∆.
In Congruent Triangles corresponding parts are always equal and we write it in short CPCT i e, corresponding parts of Congruent Triangles.
It is necessary to write a correspondence of vertices correctly for writing the congruence of triangles in symbolic form.
Criteria for congruence of triangles:
There are 4 criteria for congruence of triangles.
SAS( side angle side):
Two Triangles are congruent if two sides and the included angle of a triangle are equal to the two sides and included angle of the the other triangle.
SSS(side side side):
Three sides of One triangle are equal to the three sides of another triangle then the two Triangles are congruent.
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We know that median bisects opposite side. Use this property and then show that given parts by using SSS and SAS congruence rule.
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Solution:
Given:
AM is the median of ∆ABC & PN is the median of ∆PQR.
AB = PQ, BC = QR & AM = PN
To Show:
(i) ΔABM ≅ ΔPQN
(ii) ΔABC ≅ ΔPQR
Proof:
Since AM & PN is the median of ∆ABC
(i) 1/2 BC = BM &
1/2QR = QN
(AM and PN are median)
Now,
BC = QR. (given)
⇒ 1/2 BC = 1/2QR
(Divide both sides by 2)
⇒ BM = QN
In ΔABM and ΔPQN,
AM = PN (Given)
AB = PQ (Given)
BM = QN (Proved above)
Therefore,
ΔABM ≅ ΔPQN
(by SSS congruence rule)
∠B = ∠Q (CPCT)
(ii) In ΔABC & ΔPQR,
AB = PQ (Given)
∠B = ∠Q(proved above in part i)
BC = QR (Given)
Therefore,
ΔABC ≅ ΔPQR
( by SAS congruence rule)
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