Let O be the centre of a circle. Let chords AB = 10 \text{ cm} and CD = 24 \text{ 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 \text{ 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 only ✓
- B. II only
- C. Both I and II
- D. Neither I nor II
Correct Answer: A. I only
Explanation
Using Pythagoras theorem on both right triangles OPA and OQC with hypotenuse r: OP^2 + 5^2 = r^2 and OQ^2 + 12^2 = r^2. Since OP + OQ = 17, solving gives OP = 12 and OQ = 5. Area of \Delta OAP = 0.5 \times 5 \times 12 = 30. Area of \Delta OCQ = 0.5 \times 12 \times 5 = 30. So I is correct. However, calculating the area of \Delta OAC using coordinates yields either 84.5 \text{ cm}^2 or 59.5 \text{ cm}^2, never 102 \text{ cm}^2. Thus II is false.
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