A circle is inscribed in a square such that the circumference of the circle touches the midpoint of each side of the square. The length of the diagonal of the square is 116 units. What is the area, in square units, of the circle?13 Mark For ReviewA) 1,682B) 3,364C) 6,728D) 13,456

Step 1: Understand the problem The problem is asking for the area of a circle that is inscribed in a square. The length of the diagonal of the square is given as 116 units.

Step 2: Use the properties of a square and circle The diagonal of a square splits it into two equal right-angled triangles. The diagonal is also the hypotenuse of these triangles. In a square, all sides are equal, so the sides of the square are the legs of the right triangle.

Step 3: Use the Pythagorean theorem The Pythagorean theorem states that in a right-angled triangle, the square of the hypotenuse is equal to the sum of the squares of the other two sides. So, if 'd' is the diagonal and 'a' is the side of the square, we have:

d^2 = a^2 + a^2 d^2 = 2a^2 a^2 = d^2 / 2

Step 4: Substitute the given values Substitute d = 116 units into the equation:

a^2 = (116)^2 / 2 a^2 = 6728 a = sqrt(6728)

Step 5: Find the radius of the circle The diameter of the circle is equal to the side of the square. So, the radius 'r' of the circle is half the side of the square:

r = a / 2 r = sqrt(6728) / 2

Step 6: Find the area of the circle The area 'A' of a circle is given by the formula:

A = πr^2

Substitute r = sqrt(6728) / 2 into the formula:

A = π * (sqrt(6728) / 2)^2 A = π * 6728 / 4 A = π * 1682

So, the area of the circle is 1682π square units. Therefore, the answer is (A) 1,682π.