Suppose that a right triangle has legs of lengths 5 cm and 9 cm. (Note that the

Question

Suppose that a right triangle has legs of lengths 5 cm and 9 cm. (Note that the legs of a right triangle are the two sides that are not the hypotenuse.) A rectangle is inscribed in this right triangle so that two sides of the rectangle lie along the legs. Find the largest possible area of such a rectangle. Suppose that a right triangle has legs of lengths 5 cm and 9 cm. (Note that the legs of a right triangle are the two sides that are not the hypotenuse.) A rectangle is inscribed in this right triangle so that two sides of the rectangle lie along the legs. Find the largest possible area of such a rectangle.

Explanation / Answer

The legs can be considered as lying along the x and y axes of an x-y coordinate system. So the vertices of the triangle are at (0,0), (5,0) and (0,9).

The equation of the hypotenuse is

(y - 0)/(x - 5) = (9 - 0)/(0 - 5)

y/(x - 5) = 9/(-5) = -9/5

y = (-9/5)(x - 5)

One corner of the rectangle will lie on this line. So, the area of the rectangle will be given by

A = x(-9/5)(x - 5) where 0 < x < 5

The maximum area will be obtained by setting dA/dx = 0

dA/dx = (-9/5)(x - 5) + x(-9/5)

= (-9/5)(2x - 5)

Setting dA/dx = 0, 2x - 5 = 0, x = 2.5

y = (-9/5)(2.5 - 5)

= (9/5)(2.5)

= 4.5

So the largest inscribed rectangle will be the one which touches the hypotenuse at (2.5, 4.5).

That is, the rectangle will be 2.5 cm long along the 4.5cm leg and 5 cm long along the 9cm leg.

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