Answer:
The area of the shaded region is 57.08 units square
Step-by-step explanation:
- Lets explain some facts in the circle and the square
- If a square inscribed in a circle, then the center of the circle is
the center of the square and the length of the diameter
of the circle equal the length of the diagonal of the square
- The center of the square is the point of the intersection of its
diagonal
- The length of the diagonal of a square = √2 the length of the side
of the square ⇒ d = √2 s
* Lets solve the problem
∵ WXYZ is a square inscribed in the circle O
∴ The diameter of circle O = the diagonal of the square
∵ WX is a side of the square WXYZ
∵ XZ is a diagonal of square WXYS
∴ XZ = √2 WX
∵ OM ⊥ XY
∵ O is the center of the circle and the square
∴ OM = 1/2 the length of the side of the square
∵ OM = 5
∴ The length of the side of the square = 2 × 5 = 10
∴ WX = 10 units
∴ XZ = √2 × 10 = 10√2 units
∵ XZ is a diameter of circle O
∴ The diameter of the circle = 10√2
∵ The radius of the circle = 1/2 diameter
∴ The radius of the circle = 1/2 × 10√2 = 5√2 units
∵ Area of the circle = πr²
∴ The area of the circle = π (5√2 )² = 50π units²
∵ The length of the side of the square is 10 units
∵ The area of the square = s²
∴ Area of the square = 10² = 100 units²
∵ Area the shaded = area circle - area square
∴ Area the shaded = 50π - 100 = 57.079 units²
* The area of the shaded region is 57.08 units square
