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Find length CD? nếu góc ABC vuông, đường tròn tiếp xúc với 2 cạnh AB, BE. Chiều dài AC=3, DE=8 TÌM CD?

We are given a diagram with the following information:

1. \( \triangle ABE \) is a right triangle with \( \angle ABC = 90^\circ \).
2. A circle is inscribed, touching sides \( AB \) and \( BE \) at points \( C \) and \( D \) respectively.
3. The lengths \( AC = 3 \) and \( DE = 8 \) are given.
4. We need to find the length of \( CD \), which is a part of the line segment passing through the circle's center.

### Solution Strategy:

In a right triangle with an inscribed circle, there are specific relationships between the sides of the triangle and the radius of the inscribed circle.

Let:
- \( AB = a \)
- \( BE = b \)
- \( AE = c \)
- \( r \) be the radius of the inscribed circle.

Using the given values:
- \( AC = 3 \)
- \( DE = 8 \)

### Step 1: Use the property of the tangents from a common point.
Since the tangents from a common point to a circle are equal in length, we have:
- \( AC = AE = 3 \)
- \( BE = DE = 8 \)

Thus, \( AB = AC + CB = 3 + 8 = 11 \).

### Step 2: Use the formula for the radius of the incircle of a right triangle.
The radius \( r \) of the incircle of a right triangle can be calculated using the formula:
\[
r = \frac{a + b - c}{2}
\]
Where \( a \), \( b \), and \( c \) are the sides of the right triangle.

Plugging in the values:
- \( a = 11 \)
- \( b = 8 \)
- \( c = 3 \)

The radius \( r \) is:
\[
r = \frac{11 + 8 - 3}{2} = \frac{16}{2} = 8
\]

### Step 3: Find the length \( CD \).
Since \( CD \) is the distance between the points where the circle touches the two sides, and it passes through the center, \( CD \) is simply equal to twice the radius of the circle.

Thus, \( CD = 2 \times r = 2 \times 8 = 16 \).

### Final Answer:
The length of \( CD \) is \( 16 \).

Would you like more details on any step? 

Here are five related questions to further explore this problem:

1. How do we derive the formula for the radius of the incircle of a right triangle?
2. What are the properties of tangents to a circle from an external point?
3. How can we use the Pythagorean theorem to find the hypotenuse in this case?
4. What would change if the triangle was not a right triangle?
5. Can we solve this problem using coordinate geometry?

**Tip:** Always remember that in right triangles with incircles, the sum of the legs minus the hypotenuse gives twice the radius of the circle.

This problem involves finding the length of CD in a right triangle with a 90° angle at B. The triangle has sides touching the inscribed circle at AC = 3 and DE = 8. Learn how to solve for CD using the incircle radius formula.

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