If circle touches all the sides of a quadrilateral ABCD, if AB = 7 cm, CD = 3 cm, BC = 8.5 cm. Find the length of AD.

Answer

To find the length of side AD in the quadrilateral ABCD where a circle touches all sides, we can use the property of tangential quadrilaterals. The sum of the lengths of opposite sides of a tangential quadrilateral is equal.

Given:
- AB = 7 cm
- CD = 3 cm
- BC = 8.5 cm
- Let AD = x cm (this is what we need to find)

According to the property of tangential quadrilaterals:
AB + CD = BC + AD

Substituting the known values into the equation:
7 cm + 3 cm = 8.5 cm + x cm

Now, simplify the left side:
10 cm = 8.5 cm + x cm

Next, isolate x by subtracting 8.5 cm from both sides:
10 cm - 8.5 cm = x cm
1.5 cm = x cm

Thus, the length of AD is:
AD = 1.5 cm

### Summary of Steps:
1. Use the property of tangential quadrilaterals: AB + CD = BC + AD.
2. Substitute the known values into the equation.
3. Simplify the equation to isolate AD.
4. Solve for AD.