Mathematics — Standard (041) — CBSE Class 10 board question

Given: XY || QR, PQ/XQ = 7/3, PR = 6.3 cm

Since XY || QR, by Basic Proportionality Theorem (Thales' Theorem):

$$\frac{PX}{XQ} = \frac{PY}{YR}$$

Now, $\dfrac{PQ}{XQ} = \dfrac{7}{3}$, so $\dfrac{PX + XQ}{XQ} = \dfrac{7}{3}$, which gives $\dfrac{PX}{XQ} = \dfrac{4}{3}$

Therefore, $\dfrac{PY}{YR} = \dfrac{4}{3}$

Also, PR = PY + YR = 6.3 cm

$$\frac{PY}{YR} = \frac{4}{3} \Rightarrow PY = \frac{4}{7} \times 6.3 = 3.6 \text{ cm}$$

$$YR = 6.3 - 3.6 = \boxed{2.7 \text{ cm}}$$

Source: Chapter 6, Section 6.3 (Basic Proportionality Theorem)

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• The key theorem here is Theorem 6.1 (BPT/Thales' Theorem): if XY || QR, then PX/XQ = PY/YR.
• Since PQ/XQ = 7/3, subtract 1 from both sides (i.e., PQ = PX + XQ) to get PX/XQ = 4/3.
• Then use PY/YR = 4/3 and PY + YR = PR = 6.3 cm to find YR = 2.7 cm.
• Examiners want to see the theorem stated, the ratio set up correctly, and clear arithmetic.