mathschallenge.net

Problem

A rectangle and an equilateral triangle with equal area are inscribed in a circle with radius, 2.

Find the dimensions of the rectangle.

Solution

By drawing a diagonal in the rectangle, we form a diameter. In the case of the triangle, we use the geometrical result which states that the centre of an inscribed equilateral triangle is 1/3 the height of the triangle. Hence the radius is 2/3 the height of the triangle, and we deduce that the height of the triangle is 3.

Applying the Pythagorean theorem to the rectangle, x² + y² = 16   [1].

Similarly with the triangle, 4a² = a² + 9, so a = 3.

Hence the area of the triangle is 33.

As the areas are equal, the area of the rectangle, xy = 33,
so x²y² = 27   [2].

From [1] we get, y² = 16 x².

Substituting this into [2] we get, x²(16 x²) = 27.

Leading to the quartic, x⁴ 16x² + 27 = 0.

Solving this as a quadratic in x² we obtain the roots, x² = 837.

Using y² = 16 x² we observe that y² = 16 (837).
That is, when x² = 8 + 37, y² = 8 37, and vice versa.

Therefore the rectangle has dimensions, (8 + 37) by (8 37).

Can you generalise for a circle with radius, r?

Find the side length of the triangle if the rectangle measures, x by y.

What if the triangle has side length, a?