Question

In the figure shown, square WXYZ is inscribed in circle o. Also OM is perpendicular to XY and OM = 5. Find the area of the shaded region

Answer 1

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