# Puzzles/Geometric Puzzles/Rectangle and Circle/Solution

30 inches

## Tools Used to Solve

Pythagorean Theorem

## Solution

Given:

$x = 12 in$
$y = 6 in$

Find:

$r = ?$

We realize that every point on a circle is equidistant from the origin. This implies that the rectangle's corner touching the circle must be r inches away from the origin, where r is the radius. We can then draw a right triangle with sides $r - y$ and $r - x.$ Now apply the Pythagorean Theorem to this triangle and solve for r.

$(r-x)^2 + (r-y)^2 = r^2$
$r^2 - 2r(x+y) + x^2 + y^2 = 0$

The above equation is quadratic and can be solved by applying the Quadratic Formula.

$r=\frac{2(x+y) \pm \sqrt {(4(x+y)^2-4(x^2+y^2)\ }}{2}$

Which simplifies to,

$r=(x+y) \pm \sqrt {2xy}$

Now we can plug in our numbers and solve,

$r=(6+12) \pm \sqrt {2(6)(12)}$
$r=18 \pm 12$
$r=30$ or $r=6$

Thus the radius of our circle is 30 inches. Notice that 6 inches is not a valid answer. Why?

Comment: Simply making a statement that "we can then draw..." is confusing. Need more explanation on why r-x and r-y are valid descriptors for the triangle sides. This is easier to see with the r-x side than with the r-y.