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?

Answer: Option B

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