Question

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.

Answer 1

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!!!!