in PQR , PM=15 , PQ=25 ,PR=20, NR=8 state whether line NM is parallel to side RQ GIVE REASON
Answers
Answer:
Line NM ∥ side RQ.
Step-by-step-explanation:
NOTE: Refer to the attachment for the diagram.
We have given that,
PM = 15,
PQ = 25,
PR = 20,
NR = 8
We have to find whether line NM is parallel to side RQ or not.
From figure, we know that,
PM + MQ = PQ - - [ P - M - Q ]
⇒ 15 + MQ = 25
⇒ MQ = 25 - 15
⇒ MQ = 10
Now,
PN + NR = PR - - [ P - N - R ]
⇒ PN + 8 = 20
⇒ PN = 20 - 8
⇒ PN = 12
Now, we have to find the ratios of PN : NR & PM : MQ.
PN / NR = 12 / 8
⇒ PN / NR = 3 / 2
⇒ PN : NR = 3 : 2 - - ( 1 )
Now,
PM / MQ = 15 / 10
⇒ PM / MQ = 3 / 2
⇒ PM : MQ = 3 : 2 - - ( 2 )
Now, in △PRQ,
PN / NR = PM / MQ - - [ From ( 1 ) & ( 2 ) ]
∴ Line NM ∥ side RQ - - [ Converse of basic proportionality theorem ]
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Additional Information:
1. Basic Proportionality Theorem:
1. This theorem is related to the sides of a triangle and a line parallel to a side.
2. This theorem says that,
In a triangle, if a line is parallel to any of three sides, then that line divides the other two sides in the equal ratios.
3. It is also known as BPT in short form.
2. Converse of BPT:
1. This theorem is used to show if a line is parallel to any one of the sides of the triangle.
2. This theorem say that,
If a line divides any two sides of a triangle in equal ratios, then line is parallel to the third side.
Answer:
Given:
PM = 15,
PQ = 25,
PR = 20 and NR = 8
Now, PN = PR − NR
= 20 − 8
= 12
Also, MQ = PQ − PM
= 25 − 15
= 10
In △PRQ, PR/NR=12/8
=3/2
Also,PM/MQ=15/10
=3/2
∴PR/NR=PM/MQ
By converse of basic proportionality theorem, NM is parallel to side RQ or NM || RQ.