The figure shows a right-angled triangle ABC. D is a point on AB such that CD AB. It is given that AC = 20 cm and CD = 10 cm. Find the length of BD.


To solve this problem, we need to analyze the right-angled triangle \( \triangle ABC \) with the given conditions:

- \( AC = 20 \) cm (AC is a side of the right-angled triangle, let's assume it's the base).
- \( CD = 10 \) cm (CD is perpendicular to AB, forming another right-angled triangle \( \triangle BCD \)).

**Steps to find the length of \( BD \):**

1. **Interpret the problem:**

   - Since \( CD \) is perpendicular to \( AB \), \( D \) is the point on \( AB \) such that \( CD \) is a height from \( C \) to \( AB \). This creates two right-angled triangles: \( \triangle ACD \) and \( \triangle BCD \).

2. **Use properties of similar triangles:**

   \( \triangle ACD \) and \( \triangle BCD \) share the angle \( \angle ACD \), and both are right triangles, meaning they are similar by AA (Angle-Angle) similarity. 

3. **Set up the ratio using similar triangles:**

   - Since \( \triangle ACD \sim \triangle BCD \), the sides of the triangles are proportional.

   - Let \( AD = x \) and \( BD = y \). So, \( AB = AD + BD = x + y \).

4. **Using the similarity ratios:**

   From the similarity, we have:
   \[
   \frac{AD}{AC} = \frac{CD}{BD}
   \]

   Substituting the known values:
   \[
   \frac{x}{20} = \frac{10}{y}
   \]

   Cross-multiplying gives:
   \[
   xy = 200
   \]

   Additionally, since \( D \) divides \( AB \) into \( AD \) and \( BD \), and using the triangle similarity ratio \( \frac{AD}{AC} = \frac{CD}{BD} \), we have \( x = \frac{10}{y} \times 20 \).

5. **Solving for \( BD \):**

   We use the equation \( xy = 200 \) and let \( AD = x \), \( BD = y \). 

   Since \( AB = x + y \) and knowing the triangles' proportions, from \( \triangle ABC \), we can use:

   \[
   AC^2 = AD \cdot AB
   \]

   \[
   20^2 = x \cdot (x + y)
   \]

   We know \( xy = 200 \), and \( y = \frac{200}{x} \). Substituting this into \( AB = x + y \):

   \[
   400 = x \cdot (x + \frac{200}{x})
   \]

   Simplify:

   \[
   400 = x^2 + 200
   \]

   \[
   x^2 = 200
   \]

   \[
   x = \sqrt{200}
   \]

   Since \( x = 10\sqrt{2} \), we can find \( BD = y \):

   \[
   y = \frac{200}{x} = \frac{200}{10\sqrt{2}} = 10\sqrt{2}
   \]

   Therefore, \( BD = 10 \sqrt{2} \approx 14.14 \) cm.

**Conclusion:**
The length of \( BD \) is \( 10 \sqrt{2} \) cm.

Would you like further details, or do you have any other questions?

Here are some related questions that you might find interesting:

1. How can we calculate the area of \( \triangle ABC \)?
2. What are the lengths of \( AD \) and \( CD \) using trigonometry?
3. How would the length of \( BD \) change if \( CD \) was 15 cm instead of 10 cm?
4. How can we find the hypotenuse \( AB \) using the lengths we have?
5. What is the length of \( AB \) directly using the Pythagorean theorem?

**Tip**: When dealing with similar triangles, remember that corresponding side lengths are proportional. This property is helpful in finding unknown sides.

Learn how to find the length of BD in triangle ABC with given dimensions AC = 20 cm and CD = 10 cm. This problem involves applying the similarity of triangles and the Pythagorean theorem. Suitable for students in Grades 9-10.

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