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If CD and DA are natural numbers, find the perimeter of triangle ACD.

We are given a right-angled triangle \(ABC\) inside a larger triangle \(ACD\), and the lengths \(AB = 5\) and \(BC = 12\). We need to find the perimeter of triangle \(ACD\), where \(CD = x\) and \(DA = y\) are both natural numbers.

### Step 1: Find \(AC\)
First, apply the Pythagorean theorem to the right triangle \(ABC\):

\[
AC^2 = AB^2 + BC^2
\]

Substitute the given values:

\[
AC^2 = 5^2 + 12^2 = 25 + 144 = 169
\]

So,

\[
AC = \sqrt{169} = 13
\]

### Step 2: Perimeter of triangle \(ACD\)
The perimeter of triangle \(ACD\) is the sum of its three sides:

\[
\text{Perimeter of } \triangle ACD = AC + CD + DA = 13 + x + y
\]

### Step 3: Find natural values of \(x = CD\) and \(y = DA\)
From the diagram, it appears that \(CD = x\) and \(DA = y\) form a right triangle or at least adhere to natural number constraints. We need to find appropriate values for \(x\) and \(y\) that fit the conditions. Since \(x\) and \(y\) are natural numbers, this suggests that a relation such as a Pythagorean triple might apply here (though it's not immediately obvious from the diagram). 

Without additional specific information regarding angles or further constraints, the natural choice for \(x\) and \(y\) could be based on typical integer solutions that satisfy common geometric properties (if not directly evident from the problem's constraints).

However, based on the information given, the perimeter will depend on the specific values of \(x\) and \(y\) that the problem hints toward, likely fitting a known pattern in geometric number theory.

Would you like more detailed solutions or have specific questions about this? 

### Related Questions:
1. What are the conditions for \(x\) and \(y\) to form a valid triangle in such scenarios?
2. How does one apply the Pythagorean theorem in complex geometric shapes?
3. What are the properties of natural numbers in geometric contexts?
4. Can the diagram's angles help deduce the possible values of \(x\) and \(y\)?
5. How does this example extend to non-right-angled triangles?

**Tip:** Always check if a triangle is right-angled when working with geometric problems as it simplifies calculations with the Pythagorean theorem.

Learn how to find the perimeter of triangle ACD, where CD and DA are natural numbers, using the Pythagorean Theorem. Includes step-by-step explanations for determining the length of AC and calculating the perimeter. Suitable for Grades 8-10.

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