ABCD is a cyclic quadrilateral in which AB = 15 cm, BC = 12 cm and CD = 10 cm. If AC bisects BD, then what is the measure of AD?

Options

A. 15 cm
B. 13.5 cm
C. 18 cm
D. 20 cm

Answer

To solve the problem, we will use the property of cyclic quadrilaterals which states that the product of the lengths of the opposite sides is equal when a diagonal bisects the other diagonal.

### Step-by-Step Solution:

1. **Identify the Given Values:**
- \( AB = 15 \, \text{cm} \)
- \( BC = 12 \, \text{cm} \)
- \( CD = 10 \, \text{cm} \)
- We need to find \( AD \).

2. **Use the Property of Cyclic Quadrilaterals:**
- According to the property, if \( AC \) bisects \( BD \), then:
\[ AB \times BC = CD \times AD \]

3. **Substitute the Known Values:**
- Plugging in the values we have:
\[ 15 \times 12 = 10 \times AD \]

4. **Calculate the Left Side:**
- Calculate \( 15 \times 12 \):
\[ 15 \times 12 = 180 \]

5. **Set Up the Equation:**
- Now we have:
\[ 180 = 10 \times AD \]

6. **Solve for \( AD \):**
- To find \( AD \), divide both sides by 10:
\[ AD = \frac{180}{10} = 18 \, \text{cm} \]

7. **Final Answer:**
- Therefore, the measure of \( AD \) is \( 18 \, \text{cm} \).