Mathematics — CBSE Class 10 board question

Answer: (D) 5.25 cm

By BPT (Thales' Theorem), since PQ ∥ XY ∥ BC:

$$\frac{AP}{PX} = \frac{AQ}{QY} \Rightarrow \frac{2}{1.5} = \frac{AQ}{0.75} \Rightarrow AQ = 1 \text{ cm}$$

For CY, using $\frac{AX}{XB} = \frac{AY}{YC}$: AX = AP + PX = 3.5 cm, XB = 4 cm.

$$\frac{AY}{YC} = \frac{3.5}{4} \Rightarrow AY = AQ + QY = 1.75 \text{ cm} \Rightarrow CY = \frac{1.75 \times 4}{3.5} = 2 \text{ cm}$$

Wait — let me recheck: $\frac{AX}{XB}=\frac{AY}{YC} \Rightarrow \frac{3.5}{4}=\frac{1.75}{YC} \Rightarrow YC = \frac{1.75\times4}{3.5}=2$ cm

$$AQ + CY = 1 + 2 = \boxed{3 \text{ cm}}$$

Correct answer: (C) 3 cm

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• Apply BPT to triangle ABC with PQ ∥ BC: $\dfrac{AP}{PX} = \dfrac{AQ}{QY}$ gives AQ = 1 cm.
• Apply BPT again with XY ∥ BC: $\dfrac{AX}{XB} = \dfrac{AY}{YC}$, where AX = 2 + 1.5 = 3.5 cm, AY = AQ + QY = 1.75 cm, XB = 4 cm → CY = 2 cm.
• AQ + CY = 1 + 2 = 3 cm. Examiners expect clear ratio setup using BPT (Theorem 6.1) at each step.