circles and triangles and angles?

Question

can someone help me with this problem?
A rectangle is inscribed in a circle of radius 2007. If the diagonals form a forty-five degree angle, then what is the area of the rectangle?
plz show work. Thanks.

Answer 1

That means the rectangle can be broken into four isosceles triangles determined by the diagonals; two have apex angle 45, the other two 135. For all four, the equal side lengths are 2007. There's a nice formula for the area of isosceles triangles in terms of apex angle x and equal side length r: the area is (1/2) r^2 sin(x).

So we have 2*(1/2) 2007^2 sin(45) + 2*(1/2) 2007^2 sin(135). Since sin(45) = sin(135) ~ .7071, that gives 2007^2(.7071+.7071) = 5969521 square units.