Maximum area

Geometry Level 3

There are three concentric circles of radius 2 2 , 13 \sqrt{13} and 5 5 units. One point is selected on each circle and triangle is formed by joining these points. What is the maximum possible area of the triangle?

13.856 14.624 15.588 17.320

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Sahil Bansal by Sahil Bansal , 23, India

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1 solution

Sahil Bansal
Jun 2, 2017

Let A be the point on the innermost circle and B be the point on the middle circle.For area to be maximum for the given AB, C point should be such that the perpendicular dropped from C to AB passes through center O(to maximize the height).Now we need to find h h for which the area will be maximum.

Area of triangle= A A = 1 2 \frac{1}{2} × \times [ 4 − h 2 + 13 − h 2 \sqrt{4 - h^2} + \sqrt{13 - h^2} ] [ 5 + h 5 + h ]

Putting d A d h \frac{dA}{dh} = 0 and solving, we get h = 1 h = 1 .

Hence A A = 1 2 \frac{1}{2} × \times 3 3 3 \sqrt{3} × \times 6 6 = 15.588

4 Helpful 0 Interesting 0 Brilliant 0 Confused

A point of interest: Suppose that we place point B to the left so it is on the outer circle, and point A on the circle with radius sqrt(13), directly below the current figure. Then the new area will be 15.588. Ed Gray

Edwin Gray - 2 years, 4 months ago

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