**Quadrilateral ABCD**

Given the quadrilateral ABCD with the following measurements:

AB = 3 cm
BC = 4 cm
CD = 4 cm
DA = 5 cm
AC = 5 cm

We are required to find the area of this quadrilateral.

**Using the Formula**

To find the area of a quadrilateral, we can use the formula:

Area = ½ * diagonal1 * diagonal2 * sin(angle between diagonals)

In this case, the diagonals are AC and BD, and we need to find the angle between them.

**Finding the Angle**

To find the angle between the diagonals, we can use the Law of Cosines.

Let's consider triangle ABC. Using the Law of Cosines, we have:

AC² = AB² + BC² - 2 * AB * BC * cos(angle ABC)

Substituting the given values, we get:

5² = 3² + 4² - 2 * 3 * 4 * cos(angle ABC)

25 = 9 + 16 - 24 * cos(angle ABC)

25 = 25 - 24 * cos(angle ABC)

24 * cos(angle ABC) = 0

cos(angle ABC) = 0

angle ABC = 90 degrees

Therefore, angle ABC is a right angle.

Similarly, we can find that angle ADC is also a right angle.

**Calculating the Area**

Now, we have a rectangle ABCD with diagonal AC and BD.

The formula for the area of a rectangle is:

Area = length * width

In this case, the length is AC, which is 5 cm, and the width is BD, which we need to find.

Using the Pythagorean theorem, we can find the length of BD:

BD² = AB² + AD²

BD² = 3² + 5²

BD² = 9 + 25

BD² = 34

BD = √34 cm

Now, we can calculate the area of the rectangle:

Area = AC * BD

Area = 5 cm * √34 cm

Area = 5√34 cm²

Therefore, the area of quadrilateral ABCD is 5√34 square centimeters.