In `DeltaABC,angleB=90^(@)andBDbotAC`. If DC=7 cm and AD = 3 cm , then the length of BD is

Options

A. `sqrt(23)cm`
B. `sqrt(21)cm`
C. `sqrt(7)cm`
D. 21 cm

Answer

To solve the problem, we need to find the length of BD in triangle ABC where angle B is 90 degrees, and BD is perpendicular to AC. Given that DC = 7 cm and AD = 3 cm, we can follow these steps:

### Step-by-Step Solution:

1. **Identify the lengths**:
- We know that DC = 7 cm and AD = 3 cm.
- Therefore, the total length of AC can be calculated as:
\[ AC = AD + DC = 3 \, \text{cm} + 7 \, \text{cm} = 10 \, \text{cm} \]

2. **Apply the Pythagorean theorem in triangle BDC**:
- In triangle BDC, we can apply the Pythagorean theorem:
\[ BD^2 + DC^2 = BC^2 \]
- Let BD = x. Then, substituting the known value of DC:
\[ x^2 + 7^2 = BC^2 \quad \text{(1)} \]

3. **Apply the Pythagorean theorem in triangle ABD**:
- In triangle ABD, we also apply the Pythagorean theorem:
\[ BD^2 + AD^2 = AB^2 \]
- Again, substituting the known value of AD:
\[ x^2 + 3^2 = AB^2 \quad \text{(2)} \]

4. **Apply the Pythagorean theorem in triangle ABC**:
- For the larger triangle ABC, we have:
\[ AB^2 + BC^2 = AC^2 \]
- Substituting the known value of AC:
\[ AB^2 + BC^2 = 10^2 \quad \text{(3)} \]

5. **Substituting equations (1) and (2) into (3)**:
- From equation (1), we have \( BC^2 = x^2 + 49 \).
- From equation (2), we have \( AB^2 = x^2 + 9 \).
- Now substituting these into equation (3):
\[ (x^2 + 9) + (x^2 + 49) = 100 \]
- Simplifying this gives:
\[ 2x^2 + 58 = 100 \]

6. **Solving for x**:
- Rearranging the equation:
\[ 2x^2 = 100 - 58 \]
\[ 2x^2 = 42 \]
\[ x^2 = 21 \]
\[ x = \sqrt{21} \, \text{cm} \]

7. **Conclusion**:
- Therefore, the length of BD is:
\[ BD = \sqrt{21} \, \text{cm} \]