A triangle is drawn with its vertices on the circle C such that one of its sides is a diameter of C and the other two sides have their lengths in the ratio a:b. If the radius of the circle is r, then the area of the triangle is:

Options

1. abr²/(2(a²+b²))
2. abr²/(a²+b²)
3. 4abr²/(a²+b²)
4. 2abr²/(a²+b²)

Correct Answer

2abr²/(a²+b²)

Explanation

Since one side is diameter, the triangle is right-angled (Thales theorem). Let the two sides be ak and bk. By Pythagoras: (ak)² + (bk)² = (2r)². Solve for k, then area = (1/2) × ak × bk = 2abr²/(a²+b²).

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