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Triangles
RD Sharma
CBSE
Grade 10
Question:
In the given figure, $A, B$ and $C$ are points on $OP, OQ$ and $OR$ respectively such that $AB \parallel PQ$ and $AC \parallel PR$. Show that $BC \parallel QR$.
Step-by-Step Solution
Key Concept: Apply BPT in $\Delta OPQ$ ($AB \parallel PQ \Rightarrow \dfrac{OA}{AP} = \dfrac{OB}{BQ}$) and in $\Delta OPR$ ($AC \parallel PR \Rightarrow \dfrac{OA}{AP} = \dfrac{OC}{CR}$). Equate ratios $\Rightarrow \dfrac{OB}{BQ} = \dfrac{OC}{CR} \Rightarrow BC \parallel QR$ by converse of BPT.
In $\Delta OPQ$, $AB \parallel PQ \Rightarrow \dfrac{OA}{AP} = \dfrac{OB}{BQ}$ (by BPT). (1) [1.0 Mark] In $\Delta OPR$, $AC \parallel PR \Rightarrow \dfrac{OA}{AP} = \dfrac{OC}{CR}$ (by BPT). (2) [1.0 Mark] From (1) and (2): $\dfrac{OB}{BQ} = \dfrac{OC}{CR}$. By converse of BPT in $\Delta OQR$, $BC \parallel QR$. Proved! [1.0 Mark]
--- 🎯 Official CBSE Marking Scheme: Applying BPT in $\Delta OPQ \Rightarrow OA/AP = OB/BQ$: 1.0 Mark Applying BPT in $\Delta OPR \Rightarrow OA/AP = OC/CR$: 1.0 Mark Equating ratios and applying converse of BPT to show $BC \parallel QR$: 1.0 Mark
Correct Answer:
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