In the given figure,PQRS is a parallelogram and L is the mid-point of RQ.Prove that M is the point of trisection of PR and SL.
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In the given figure,PQRS is a parallelogram and L is the mid-point of RQ.Prove that M is the point of trisection of PR and SL.
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Hi,
Answer:
It is given that PQRS is a parallelogram and opposite facing sides of a //gm are of equal length.
∴ SP = RQ & SR = PQ …… (i)
Also, given L is the midpoint of RQ
∴ RL = LQ = ½ * RQ ….. (ii)
Step 1:
Let’s consider ∆ PSM & ∆ RLM,
∠SMP = ∠RML ….. [vertically opposite angles of //gm PQRS]
∠SPM = ∠MRL ….. [alternate angles of //gm PQRS]
∴ By AA similarity, ∆ PSM ~ ∆ RLM
Since corresponding sides of two similar triangles are proportional to each other.
∴ SP/RL = SM/LM = PM/RM ….. (iii)
Step 2:
From (i) & (ii), we get
RL = ½ SP ….. (iv)
From (iii) & (iv), we get
SP/(SP/2) = PM/RM
⇒ 2SP/SP = PM/RM
⇒ PM = 2 RM ….. (v)
Now, we can also write
PR = PM + RM
⇒ PR = 2RM + RM …. [substituting from (v)]
⇒ PR = 3 RM
⇒ RM = 1/3 * PR
Hence, M is a point of trisection of PR.
Step 3:
Similarly,
From (iii) & (v), we get
SM/LM = 2RM/RM
⇒ SM = 2 LM …. (vi)
Now, we can also write
SL = SM + LM
⇒ SL = 2LM + LM …. [substituting from (vi)]
⇒ SL = 3 LM
⇒ LM = 1/3 * SL
Hence, M is a point of trisection of SL.
Thus, it is proved that point M is a trisection of both PR & SL.
Hope this is helpful!!!!