CDS 2026 (I) Elementary Mathematics, question 65

Topic: Geometry

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Q65

Let O be the centre of a circle. Let the chords $AB=10$ cm and $CD=24$ cm be two parallel chords of the circle. The two chords are on opposite side of the centre and the distance between them is $17$ cm. Let P be the midpoint of AB and Q be the midpoint of CD. Further, the points O and A, the points O and C, the points A and C are joined. Which of the statements given below is/are correct ?

I. Area of triangle OAP is equal to area of triangle OCQ.

II. Area of triangle OAC is equal to $102\text{ cm}^{2}$ .

Select the answer using the code given below :

(a) I onlyCorrect answer
(b) II only
(c) Both I and II
(d) Neither I nor II

Solution.

Answer: (a) I only

Let $OP = u$ and $OQ = v$ . Since the chords are on opposite sides, $u + v = 17$ . Equal radii give $u^{2} + 5^{2} = v^{2} + 12^{2}$ , so $(u - v)17 = 119$ .
Hence $u = 12,v = 5$ and the circle radius is 13 cm. The two right-triangle areas are $(1/2) \times 5 \times 12 = 30{cm}^{2}$ and $(1/2) \times 12 \times 5 = 30{cm}^{2}$ .
Thus I is true. For II, any triangle OAC formed by two radii has area at most $(1/2) \times OA \times OC = 169/2 = 84.5{cm}^{2}$ .
Therefore its area cannot be $102{cm}^{2}$ .
Shortcut: An area bound settles II without choosing which endpoints are A and C.