Puzzles/Geometric Puzzles/Rectangle and Circle/Solution

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Answer[edit]

30 inches

Tools Used to Solve[edit]

Pythagorean Theorem

Quadratic Formula

Solution[edit]

RectangleAndCircleSolution.jpg

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.