The circular base of an above-ground swimming pool lies in a level yard and just touches two straight sides of a fence at points A and B, as shown in the figure above. Point C is on the ground where the two sides of the fence meet. How far from the center of the pool's base is point A?Consider the diagram below:
Notice that we are asked to find the length of QA, or the radius of the circular base.
(1) The base has area 250 square feet --> \(area=\pi{r^2}=250\) --> we can find r. Sufficient.
(2) The center of the base is 20 feet from point C. Triangle CQA IS a right triangle, because the tangent line (CA) is always at the 90 degree angle (perpendicular) to the radius (QA) of a circle.
So, we have that CQ=hypotenuse=20.
BUT, knowing that hypotenuse equals to 20 DOES NOT mean that the sides of the right triangle necessarily must be in the ratio of Pythagorean triple - 12:16:20. Or in other words: if \(x^2+y^2=20^2\) DOES NOT mean that \(x=12\) and \(y=16\). Certainly this is one of the possibilities but definitely not the only one. In fact \(x^2+y^2=20^2\) has infinitely many solutions for \(x\) and \(y\) and only one of them is \(x=12\) and \(y=16\).
For example: \(x=1\) and \(y=\sqrt{399}\) or \(x=2\) and \(y=\sqrt{396}\)...
So, this statement is not sufficient to get QA.
Answer: A.
For the sake of argument(I know that this doesn't provide any additional value to help solve this problem) -- how can we assume that BCQ and ACQ are equal? I realize that both lines are tangent to the circle but doesn't that mean that the line is perpendicular to the center of the circle. Theoretically, that has infinite points, doesn't it?