Rectangle ABCD is inscribed in circle P. What is the area of circle P?

Question

https://gmatclub.com/forum/download/file.php?id=21438 Rectangle ABCD is inscribed in circle P. What is the area of circle P? (1) The area of rectangle ABCD is 100. (2) Rectangle ABCD is a square. inscribedsquare.jpg

AccipiterQ · Answer

1. area of rectangle ABCD is 100. If the rectangle is also a square, then you can solve. It doesn't say that though, so the rectangle could have a number of different side lengths. insufficient.

2. rectangle ABCD is a square. No values are given. Insufficient.

1&2 combined. the rectangle is also a square, meaning that all sides are length of 10. you can draw a line from a to c (which would be the cirlce's diameter), and form a right triangle, with two sides of 10, and AC (the hypothaneuse (sp??) being unknown).

10^2+10^2=X^2
200=X^2
X= \sqrt{200} 
X=10* \sqrt{2} 
so the diameter is 10* \sqrt{2}

since the radius is half the diamter you get 5* \sqrt{2}

area of a cirlce is pi*r^2, so pi*(5* \sqrt{2} )^2
=pi*25*2
=50*pi

so 1&2 are sufficient.

Bunuel · Answer

When combined: the area of a square is (diagonal)^2/2, so we are given that (diagonal)^2/2=100 -->  d=10\sqrt{2} .