The correct answer is option (B):
27 units
Let the sides of the quadrilateral be a, b, c, and d. We are given that the perimeter is 126 cm, so a + b + c + d = 126. We are also given that three of the sides are in the ratio 5:8:9. Let's assume a:b:c = 5:8:9. This means we can write a = 5x, b = 8x, and c = 9x for some value of x.
Substituting these into the perimeter equation, we get 5x + 8x + 9x + d = 126. This simplifies to 22x + d = 126.
Now, a key property of quadrilaterals circumscribed about a circle is that the sum of opposite sides are equal. In other words, a + c = b + d. Substituting our expressions, we have 5x + 9x = 8x + d. This simplifies to 14x = 8x + d. Subtracting 8x from both sides gives us 6x = d.
Now we can substitute d = 6x into the perimeter equation 22x + d = 126. This becomes 22x + 6x = 126, which simplifies to 28x = 126. Dividing both sides by 28, we get x = 126/28 = 4.5.
Finally, we can find the length of the fourth side, d, by substituting the value of x we just calculated: d = 6x = 6 * 4.5 = 27. Therefore, the length of the fourth side is 27 cm.