Puzzles/Geometric Puzzles/Rectangle and Circle/Solution

Answer
30 inches

Tools Used to Solve
Pythagorean Theorem

Quadratic Formula

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.