The diagonals of a quadrilateral ABCD intersect each other at the point O such that AO/BO = CO/OD. Show that quadrilateral ABCD is a trapezium.

MathsCBSE Class 10 · Basic · Board exam 2023, set 430/1/1 · 3 marks · Short answer

CBSE Class 10 Mathematics Basic 2023: question 31(b)

Set in the CBSE Class 10 board exam of 2023, paper 430/1/1, section C, question 31(b), for 3 marks. Try it first; the hints and our step-by-step solution are below.

Q31(b) · 3 marks · Short answerTriangles

The diagonals of a quadrilateral ABCD intersect each other at the point O such that $\frac{A O}{B O} = \frac{C O}{O D}$ . Show that quadrilateral ABCD is a trapezium.

Figure in the paper: Quadrilateral ABCD with AB at the bottom and DC at the top, DC longer than AB. Diagonals AC and BD intersect at O. See it in the official paper.

Source: official question paper from cbse.gov.in (PDF). See it in the full 2023 board paper, set 430/1/1.

Hints

Open one hint at a time and try again before the next.

Hint 1

Rearrange the given ratio so it compares sides of triangles AOB and COD.

Hint 2

Similar triangles give equal angles; check whether they are alternate angles for AB and CD.

Step-by-step solution

Each step shows the marks it earns, as an examiner would award them. Checked by an independent re-solve from the question paper.

Given $\frac{A O}{B O} = \frac{C O}{D O} \Rightarrow \frac{O A}{O C} = \frac{O B}{O D}$ .
½ mark
In $△ A O B$ and $△ C O D$ : $\frac{O A}{O C} = \frac{O B}{O D}$ and $∠ A O B = ∠ C O D$ [vertically opposite angles]. So $△ A O B \sim △ C O D$ [SAS similarity].
1½ marks
Hence $∠ O A B = ∠ O C D$ , i.e. $∠ C A B = ∠ A C D$ [corresponding angles of similar triangles]. These are alternate interior angles for lines AB and CD with transversal AC, so $A B ∥ C D$ . Therefore ABCD is a trapezium.
1 mark

Answer: AB ∥ DC, so ABCD is a trapezium

Where the marks are

Rearranges the given ratio to $\frac{O A}{O C} = \frac{O B}{O D}$ and uses vertically opposite angles (part of 2 marks)
Proves $△ A O B \sim △ C O D$ by SAS (2 marks total)
Uses equal alternate angles to get $A B ∥ C D$ and conclude trapezium (1 mark)

Revise the chapter

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

Practise this chapterBoard-style questions from Triangles, one at a time, marked against the answer key and marking points.Back to the full paper2023 board paper, set 430/1/1 (Basic): every question, by section.

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CBSE Class 10 Mathematics Basic 2023: question 31(b)

Hints

Step-by-step solution

Revise the chapter

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