Find the side AD of the quadrilateral ABCD, if AB = 3, BC = 4, CD = 5 and it is known

Find the side AD of the quadrilateral ABCD, if AB = 3, BC = 4, CD = 5 and it is known that a circle can be inscribed in the quadrilateral ABCD.

From the condition, we know that we are given a quadrilateral ABCD, and the length of its three sides AB = 3, BC = 4, CD = 5 is also known. And we also know that a circle can be inscribed in the quadrilateral ABCD.

We need to find the fourth side of the quadrilateral AD.

For this we apply the following property of a quadrangle – a circle can be inscribed in a quadrilateral if the sums of its opposite sides are equal to each other.

That is, we write down the equality AB + CD = BC + AD.

Let’s compose and solve the equation:

3 + 5 = 4 + x;

x = 4.

Answer: AD = 4.