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Puzzle #27

A circle labeled/labelled one hundred sixty nine pi is aligned with the a square's base and tangent to its top right vertex. A double arrow from the square's bottom right vertex and the rightmost point of the circle is labeled/labelled twenty five.

A circle with area 169π169\pi is aligned with the square’s base.

What is the area of the square?

Solution

As the area of the circle is 169π169\pi, its radius is 13.

Let xx be the square’s side length.

Consider the right-angled/right triangle below. Its hypotenuse is 13, its base’s length is 25−13=1225 − 13 = 12 and its height is 13−x13 − x:

A circle is aligned with a square's base and tangent to its top right vertex. The right side of the square is labeled/labelled x. A right/right-angled triangle has vertices the center/centre of the circle, the square's top right vertex and a third point vertically below the center/centre of the circle. The base of the right/right/angled triangle, its height and its hypotenuse are labeled/labelled twelve, thirteen minus x and thirteen, respectively.

Using the Pythagorean /Pythagoras’ theorem:

132=122+(13−x)213^2=12^2+\left(13-x\right)^2
169=144+169−26x+x2169=144+169-26x+x^2
x2−26x+144=0x^2-26x+144=0
(x−8)(x−18)=0\left(x-8\right)\left(x-18\right)=0
x=8,x=18x=8, \, x=18

As the square’s length is less than the radius, then

x=8x=8

So, the area of the square is

828^2
6464

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