In figure, seg PD is a median of PQR. Point T is the mid point of seg PD. Produced QT intersects PR at M. Show that $\frac{P M}{P R} = \frac{1}{3} .$ [Hint : Draw DN || QM.]

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Solution

Given: PD is a median.
T is the midpoint of PD.
QT intersects PR at M
To prove :
$\frac{P M}{P R} = \frac{1}{3}$
Proof:
In $△$ PDN,
Point T is the midpoint of seg PD
Given seg TM || seg DN [DN||QM and Q-T-M]
Point M is the midpoint of seg PN. [Construction and Q-T-M]
PM = MN ….(i) [Converse of midpoint theorem]
In $△$ QMR,
Point D is the midpoint of seg QR and seg DN || seg QM [Construction]
Point N is the midpoint of seg MR. [Converse of midpoint theorem]
MN = NR …..(ii)
PM = MN = RN …..(iii) [From (i) and (ii)]
Now, PR = PM + MN + RN [ P-M-R]
PR = PM + PM + PM [From (iii) ]
PR = 3 PM
$\frac{P M}{P R} = \frac{1}{3}$
Hence proved.

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