Question

Find the area of a cyclic quadrilateral whose two sides measure 4 and 5 units, and whose diagonal coincides with a diameter of the circle. Suppose the radius of the circumscribing circle is 2 squareroot 3 units. One diagonal of a cyclic quadrilateral coincides with a diameter of a circle whose area is 36 pi cm^2. If the other diagonal which measures 8 cm meets the first diagonal at right angles, find the area of the quadrilateral. Find the lengths of the sides of the cyclic quadrilateral in number 32.

Answer 1

The area of the cyclic quadrilateral is 16√3 square units.

Step-by-step explanation:

The area of a cyclic quadrilateral can be found by using the formula:

Area = (1/2) * product of the lengths of the diagonals * sine of the angle between the diagonals.

In this case, one diagonal coincides with the diameter of the circle. Since the radius of the circle is given as 2√3 units, the length of the diagonal is 2 times the radius, which is 4√3 units. The other diagonal is given as 8 cm, and it is perpendicular to the first diagonal. Therefore, the angle between the diagonals is 90 degrees.

Using the formula, the area of the quadrilateral is:

Area = (1/2) * (4√3)(8) * sin(90)

Area = 16√3 square units.