O is the point of intersection of two chords AB and CD of a circle. If ∠ BOC = 80° and OA = OD then △ ODA and △ OBC are (A) equilateral and similar (B) isosceles and similar (C) isosceles but not similar (D) not similar

MathsCBSE Class 10 · Standard · Sample paper, session 2024-25 · 1 mark · MCQ

CBSE Class 10 Mathematics Standard sample paper 2024-25: question 10

From the official CBSE sample question paper for 2024-25, section A, question 10, for 1 mark. Try it first; the hints and our step-by-step solution are below.

Q10 · 1 mark · MCQCircles

$O$ is the point of intersection of two chords $A B$ and $C D$ of a circle. If $∠ B O C = 8 0^{\circ}$ and $O A = O D$ then $△ O D A$ and $△ O B C$ are

(A)equilateral and similar
(B)isosceles and similar
(C)isosceles but not similar
(D)not similar

Figure in the paper: A circle with two chords AB and CD intersecting at the point O inside the circle. Points on the circle: A (bottom left), B (top right), C (bottom right), D (top left). Segment OA and OD are marked equal (tick marks). The angle BOC at O is marked. See it in the official paper.

Source: official sample paper from cbseacademic.nic.in (PDF). See it in the full Sample paper 2024-25.

Hints

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

Hint 1

Angles in the same segment of a circle are equal.

Hint 2

Compare the angles of the two triangles at O, A, D and at O, C, B.

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.

$O A = O D$ , so $△ O D A$ is isosceles and $∠ O A D = ∠ O D A$ .
$∠ A O D = ∠ B O C = 8 0^{\circ}$ [vertically opposite]. Also $∠ O A D = ∠ O C B$ and $∠ O D A = ∠ O B C$ [angles in the same segment]. So $△ O D A \sim △ O C B$ (AA).
Hence $∠ O B C = ∠ O C B$ , so $△ O B C$ is also isosceles. The base angles are $5 0^{\circ}$ each, so neither triangle is equilateral. Both are isosceles and similar: option (B).
1 mark

Answer: Isosceles and similar, option (B)

Where the marks are

Shows both triangles are isosceles and similar (AA) using vertically opposite angles and angles in the same segment (1 mark)

Revise the chapter

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

Practise this chapterBoard-style questions from Circles, one at a time, marked against the answer key and marking points.Back to the full paperSample paper 2024-25 (Standard): every question, by section.

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CBSE Class 10 Mathematics Standard sample paper 2024-25: question 10

Hints

Step-by-step solution

Revise the chapter

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