ABCD is a parallologram. From C perpendiculars CS and CT are drpopped upon AB and AD, respectively. Given `AB = 16 cm, CS = 9 cm, and CT = 8 cm, ` find BC.

Answer

To solve the problem, we need to find the length of side BC in the parallelogram ABCD, given the following information:

- AB = 16 cm
- CS (height from C to AB) = 9 cm
- CT (height from C to AD) = 8 cm

### Step-by-step Solution:

1. **Understanding the Area of the Parallelogram**:
The area of a parallelogram can be calculated using the formula:
\[ \text{Area} = \text{Base} \times \text{Height} \]

2. **Calculating the Area using Base AB**:
We can take AB as the base of the parallelogram. Therefore, the area can be calculated as:
\[ \text{Area} = AB \times CS = 16 \, \text{cm} \times 9 \, \text{cm} = 144 \, \text{cm}^2 \]

3. **Calculating the Area using Base AD**:
Now, we can also take AD as the base of the parallelogram. The area can also be expressed as:
\[ \text{Area} = AD \times CT \]
We know that CT = 8 cm, but we need to find the length of AD.

4. **Setting the Areas Equal**:
Since both expressions represent the area of the same parallelogram, we can set them equal to each other:
\[ 16 \, \text{cm} \times 9 \, \text{cm} = AD \times 8 \, \text{cm} \]
This simplifies to:
\[ 144 \, \text{cm}^2 = AD \times 8 \, \text{cm} \]

5. **Solving for AD**:
To find AD, we can rearrange the equation:
\[ AD = \frac{144 \, \text{cm}^2}{8 \, \text{cm}} = 18 \, \text{cm} \]

6. **Finding BC**:
In a parallelogram, opposite sides are equal. Therefore, since AD = 18 cm, it follows that:
\[ BC = AD = 18 \, \text{cm} \]

### Final Answer:
The length of BC is **18 cm**.

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