To find the area of a cyclic quadrilateral, we can use Brahmagupta's formula. This formula states that the area of a cyclic quadrilateral can be found by taking the square root of the product of the differences between the semiperimeter and each of the quadrilateral's sides.
In this case, we are given that the two sides of the quadrilateral measure 4 and 5 units, and the diagonal coincides with a diameter of the circle. This means that the two sides and the diameter of the circle form a right triangle. Since the radius of the circumscribing circle is given as 2β3 units, the diameter of the circle is 2 times the radius, which is 4β3 units.
Using the Pythagorean theorem, we can find the length of the remaining side of the right triangle (which is also a side of the quadrilateral). Let's call this side "x". Using the given sides of length 4, 5, and the hypotenuse of length 4β3, the Pythagorean theorem can be applied as follows:
x^2 + 4^2 = 5^2
x^2 + 16 = 25
x^2 = 9
x = 3
Now, we can find the semiperimeter of the quadrilateral by adding the lengths of all four sides and dividing by 2:
Semiperimeter = (4 + 5 + 4β3 + 3) / 2
Semiperimeter = (12 + 4β3) / 2
Semiperimeter = 6 + 2β3
Next, we can substitute the values into Brahmagupta's formula to find the area of the cyclic quadrilateral:
Area = β((Semiperimeter - 4) * (Semiperimeter - 5) * (Semiperimeter - 4β3) * (Semiperimeter - 3))
Area = β((6 + 2β3 - 4) * (6 + 2β3 - 5) * (6 + 2β3 - 4β3) * (6 + 2β3 - 3))
Area = β((2 + 2β3) * (1 + 2β3) * (2β3 + 2) * (3 + 2β3))
Area = β(4 + 4β3 + 2β3 + 12 + 4β3 + 8β3 + 4β3 + 8 + 12β3 + 6 + 6β3 + 12)
Area = β(56 + 42β3 + 12 + 18β3)
Area = β(68 + 60β3)
Finally, we can calculate the approximate value using a calculator:
Area β β(68 + 60β3)
Area β β(68) + β(60β3)
Area β 8.246
Therefore, the area of the cyclic quadrilateral is approximately 8.246 square units.