M and N are points on the sides PQ and PR respectively of a ΔPQR . For each of the following cases, state whether MN||QR :
(i) PM = 4 cm, QM = 4.5 cm, PN = 4 cm, NR = 4.5 cm
(ii) PQ = 1.28 cm, PR = 2.56 cm, PM = 0.16 cm, PN = 0.32 cm
Answers
For each of the following cases MN || QR.
Step-by-step explanation:
(i) Given : PM = 4 cm, QM = 4.5 cm, PN = 4 cm, and NR = 4.5 cm.
In ∆PQR,
PM/QM = 4/4.5
And, PN/NR = 4/4.5
so, PM/QM = PN/NR
Hence, MN || QR
[By Converse of basic proportionality theorem]
(ii) GIVEN : PQ = 1.28 cm, PR = 2.56 cm, PM = 0.16 cm, and PN = 0.32 cm.
MQ = PQ - PM
MQ = 1.28 - 0.16
MQ = 1.12
NR = PR - PN
NR = 2.56 - 0.32
NR = 2.24
In ∆PQR,
PM/QM = 0.16/1.12 = 1/7
And, PN/NR = 0.32/2.24 = 1/7
so, PM/QM = PN/NR
Hence, MN || QR
[By Converse of basic proportionality theorem
Converse of basic proportionality theorem :
If a line divides any two sides of a triangle in the same ratio then the line must be parallel to the third side.
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Step-by-step explanation:
(i) Given : PM = 4 cm, QM = 4.5 cm, PN = 4 cm, and NR = 4.5 cm.
In ∆PQR,
PM/QM = 4/4.5
And, PN/NR = 4/4.5
so, PM/QM = PN/NR
Hence, MN || to QR
[By Converse of basic proportionality theorem]
(ii) GIVEN : PQ = 1.28 cm, PR = 2.56 cm, PM = 0.16 cm, and PN = 0.32 cm.
MQ = PQ - PM
MQ = 1.28 - 0.16
MQ = 1.12
NR = PR - PN
NR = 2.56 - 0.32
NR = 2.24
In ∆PQR,
PM/QM = 0.16/1.12 = 1/7
And, PN/NR = 0.32/2.24 = 1/7
so, PM/QM = PN/NR
Hence, MN || to QR
[By Converse of basic proportionality theorem