In the given figure, OA · OB = OC · OD. Prove that △ AOD ~ △ COB.

△ AOD ~ △ COB (by SAS similarity)

MathsCBSE Class 10 · Basic · Board exam 2024, set 430/2/1 · 2 marks · Very short answer

CBSE Class 10 Mathematics Basic 2024: question 24(a)

Set in the CBSE Class 10 board exam of 2024, paper 430/2/1, section B, question 24(a), for 2 marks. Try it first; the hints and our step-by-step solution are below.

Q24(a) · 2 marks · Very short answerTriangles

In the given figure, $O A \cdot O B = O C \cdot O D$ . Prove that $△ A O D \sim △ C O B$ .

Figure in the paper: Two line segments AB and CD intersect at point O inside the figure. A and D are on one side (A upper-left, D lower-left), and C and B are on the other side (C upper-right, B lower-right). Segments AD and CB are drawn to form the two triangles AOD and COB. See it in the official paper.

Source: official question paper from cbse.gov.in (PDF). See it in the full 2024 board paper, set 430/2/1.

Hints

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

Hint 1

Turn the product equation into a ratio of sides.

Hint 2

The included angle between OA and OD, and between OC and OB, is at O. How are these two angles related?

Hint 3

Two proportional sides with an equal included angle give which similarity criterion?

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 $O A \cdot O B = O C \cdot O D$ . Dividing both sides by $O C \cdot O B$ : $\frac{O A}{O C} = \frac{O D}{O B}$ ...(i)
½ mark
In $△ A O D$ and $△ C O B$ : $\frac{O A}{O C} = \frac{O D}{O B}$ [from (i)], and $∠ A O D = ∠ C O B$ [vertically opposite angles]
1 mark
Therefore $△ A O D \sim △ C O B$ [SAS similarity criterion].
½ mark

Answer: $△ A O D \sim △ C O B$ (by SAS similarity)

Where the marks are

Rewrites the given product as the ratio $\frac{O A}{O C} = \frac{O D}{O B}$ (½ mark)
Uses the ratio and the vertically opposite angles $∠ A O D = ∠ C O B$ (1 mark)
Concludes similarity by SAS (½ mark)

Revise the chapter

Chapter: Triangles (notes, formulas, where marks are lost and every past question)

Practise this chapterBoard-style questions from Triangles, one at a time, marked against the answer key and marking points.Back to the full paper2024 board paper, set 430/2/1 (Basic): every question, by section.

The question is transcribed from the official paper; figures are described in words. The solution, hints and chapter tag are ours. Spotted an error? Write tosupport@questionbuddy.in.

CBSE Class 10 Mathematics Basic 2024: question 24(a)

Hints

Step-by-step solution

Revise the chapter

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