If in a `Delta` ABC, right angle at A a perpendicular AD is drawn from A on side BC such that AD = 8.4 cm, BD = 4.8 cm then find the length of side BC.

Options

A. (A). 18 cm
B. (B). 13.5 cm
C. (C). 14 cm
D. (D). 19.5 cm

Answer

To solve the problem, we will follow these steps:

1. **Understand the triangle configuration**: We have a right triangle ABC with a right angle at A. A perpendicular AD is drawn from A to side BC. We know the lengths of AD and BD.

2. **Identify the known values**:
- AD = 8.4 cm (the height from A to BC)
- BD = 4.8 cm (the segment of BC from B to D)

3. **Use the properties of similar triangles**: Since triangles ABD and ACD are similar (both are right triangles sharing angle A), we can set up a proportion based on the similarity of triangles.

4. **Set up the proportion**:
\[ \frac{CD}{AD} = \frac{AD}{BD} \]
Here, CD is the segment of BC from D to C, which we need to find.

5. **Substitute the known values into the proportion**:
\[ \frac{CD}{8.4} = \frac{8.4}{4.8} \]

6. **Cross-multiply to solve for CD**:
\[ CD \cdot 4.8 = 8.4 \cdot 8.4 \]
\[ CD \cdot 4.8 = 70.56 \]
\[ CD = \frac{70.56}{4.8} \]
\[ CD = 14.7 \text{ cm} \]

7. **Calculate the length of side BC**:
\[ BC = BD + CD \]
\[ BC = 4.8 + 14.7 \]
\[ BC = 19.5 \text{ cm} \]

8. **Final answer**: The length of side BC is 19.5 cm.