In `Delta` ABC, BP and CQ are perpendiculars from B and C to the opposite sides. If AB = 16 cm. AC= 24 cm, and CQ=20.4cm, find BP.

Answer

To find the length of BP in triangle ABC, we can use the relationship between the areas formed by the perpendiculars from points B and C to the opposite sides.

### Step-by-Step Solution:

1. **Identify the Given Values**:
- AB = 16 cm
- AC = 24 cm
- CQ = 20.4 cm
- We need to find BP.

2. **Understanding the Area of Triangle**:
The area of triangle ABC can be calculated using two different pairs of base and height:
- First Pair: Base = AB and Height = CQ
- Second Pair: Base = AC and Height = BP

3. **Area Formula**:
The area of a triangle is given by the formula:
\[ \text{Area} = \frac{1}{2} \times \text{Base} \times \text{Height} \]

4. **Set Up the Area Equations**:
From the first pair (AB and CQ):
\[ \text{Area} = \frac{1}{2} \times AB \times CQ = \frac{1}{2} \times 16 \times 20.4 \]

From the second pair (AC and BP):
\[ \text{Area} = \frac{1}{2} \times AC \times BP = \frac{1}{2} \times 24 \times BP \]

5. **Equate the Areas**:
Since both expressions represent the area of the same triangle, we can set them equal to each other:
\[ \frac{1}{2} \times 16 \times 20.4 = \frac{1}{2} \times 24 \times BP \]

6. **Cancel Out \(\frac{1}{2}\)**:
We can cancel \(\frac{1}{2}\) from both sides:
\[ 16 \times 20.4 = 24 \times BP \]

7. **Calculate \(16 \times 20.4\)**:
\[ 16 \times 20.4 = 326.4 \]

8. **Solve for BP**:
Now, we can solve for BP:
\[ 326.4 = 24 \times BP \]
Dividing both sides by 24:
\[ BP = \frac{326.4}{24} \]

9. **Perform the Division**:
\[ BP = 13.6 \text{ cm} \]

### Final Answer:
BP = 13.6 cm