Cyclic Quadrilateral problem

Geometry Level pending

A cyclic quadrilateral's diagonals are lengths $x+2.2$ and $x+3.43$ . It also has side lengths $x-1$ , $x$ , $x+2$ , and $x+3$ .

Determine the area of the region of the circumscribed circle that is not taken up by the quadrilateral. A Scientific Calculator is allowed.

44.84 10.67 23.45 21.55 20

1 solution

Yashas Ravi
Apr 17, 2018

Use Ptolemy's theorem and the Quadratic formula to determine the side lengths.
Use Brahmagupta's theorem to determine the area of the quadrilateral.
Use the circumradius formula to determine the radius of the circumscribed circle. Let 2 of the consecutive sides of the quadrilateral and the common diagonal that the sides intersect be the sides of the triangle. The area can be determined by squaring the radius and multiplying it by π.
Subtract the quadrilateral's area from the circle's area for the final answer of 21.55.