In a `triangleABC, angleBCA = 90^@` and CD is perendicular to AB. If AD = 4 cm and BD = 9 cm, then the value of DC will be

Answer

To solve the problem, we will use the properties of similar triangles.

### Step-by-Step Solution:

1. **Identify the Given Information:**
- Triangle ABC has angle BCA = 90°.
- CD is perpendicular to AB.
- AD = 4 cm.
- BD = 9 cm.

2. **Determine the Length of AB:**
- Since AD and BD are segments of AB, we can find the total length of AB:
\[ AB = AD + BD = 4 \, \text{cm} + 9 \, \text{cm} = 13 \, \text{cm}. \]

3. **Use the Property of Similar Triangles:**
- Since CD is perpendicular to AB, triangles BDC and CDA are similar to triangle BCA (by AA similarity criterion).
- From the similarity of triangles, we have:
\[ \frac{CD}{AD} = \frac{BD}{CD}. \]

4. **Set Up the Proportion:**
- Let DC = x. Then, we can write the proportion as:
\[ \frac{x}{4} = \frac{9}{x}. \]

5. **Cross-Multiply to Solve for x:**
- Cross-multiplying gives:
\[ x^2 = 4 \times 9. \]
- Simplifying this:
\[ x^2 = 36. \]

6. **Take the Square Root:**
- Taking the square root of both sides:
\[ x = \sqrt{36} = 6 \, \text{cm}. \]

7. **Conclusion:**
- Therefore, the length of DC is 6 cm.

### Final Answer:
DC = 6 cm.

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