In the given figure MNOP is a parallelogram. PM is extended to Z. OZ intersects MN and PN at Y and X respectively. If OX = 27 cm and XY = 18 cm, then what is the length (in cm) of YZ?

Solution (By Examveda Team)

In ΔXNY & ΔXPO,
∠YXN = ∠PXO [Vertical opposite angles]
and ∠YNX = ∠XPO [alternate angles]
ΔXNY is similar to ΔXPO
$$\eqalign{ & \Rightarrow \frac{{YN}}{{PO}} = \frac{a}{b} = \frac{{18}}{{27}} = \frac{{YX}}{{XO}} \cr & \Rightarrow \frac{a}{b} = \frac{2}{3} \cr & \therefore \frac{{MY}}{{PO}} = \frac{{MN - YN}}{{PO}} = \frac{{3 - 2}}{3} = \frac{1}{3} \cr & {\text{Now, as }}\Delta ZMY \sim \Delta ZPO \cr & \therefore \frac{{MY}}{{PO}} = \frac{{ZY}}{{ZO}} \cr & \Rightarrow \frac{1}{3} = \frac{x}{{x + 45}} \cr & \Rightarrow x + 45 = 3x \cr & \Rightarrow 2x = 45 \cr & \Rightarrow x = 22.5 \cr & \therefore ZY = 22.5{\text{ cm}} \cr} $$