CBSE Class 10 Mathematics Basic 2026: question 33

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Step-by-step solution

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1. Basic Proportionality Theorem: If a line is drawn parallel to one side of a triangle so that it meets the other two sides at distinct points, then it divides those two sides in the same ratio.

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2. Given: ABCD is a quadrilateral whose diagonals AC and BD meet at O, with $\frac{AO}{BO}=\frac{CO}{DO}$. To prove: ABCD is a trapezium. Construction: Through O, draw $OE\parallel AB$ meeting AD at E. (Figure: quadrilateral ABCD with diagonals meeting at O and the line OE drawn parallel to AB.)

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3. In $△ABD$, $EO\parallel AB$, so by BPT: $\frac{DE}{EA}=\frac{DO}{OB}$ …(1)

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4. Given $\frac{AO}{BO}=\frac{CO}{DO}\Rightarrow \frac{AO}{CO}=\frac{BO}{DO}\Rightarrow \frac{CO}{AO}=\frac{DO}{BO}$ …(2). From (1) and (2): $\frac{DE}{EA}=\frac{CO}{OA}$.

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5. In $△ADC$, E and O divide sides AD and AC in the same ratio, so $EO\parallel DC$ [converse of BPT]. Since $EO\parallel AB$ and $EO\parallel DC$, we get $AB\parallel DC$. Hence ABCD has a pair of parallel opposite sides, so ABCD is a trapezium.

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Answer: BPT stated. $AB\parallel DC$, so ABCD is a trapezium.

Revise the chapter

• Chapter: Triangles (notes, formulas, where marks are lost and every past question)
• Topic: Basic Proportionality Theorem (Thales Theorem)