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To solve for the length of \( x \) in the triangle, we can approach the problem using the **Pythagorean theorem** and **similar triangles**.

### Step-by-Step Breakdown:

#### 1. Identify the right triangles:
- Triangle \( ABC \) is a right triangle with:
  - \( AC = 5 \, \text{cm} \) (vertical leg)
  - \( AB = x \, \text{cm} \) (base)
  - Right angle at \( A \)

- Triangle \( ACD \) is another right triangle with:
  - \( AD = 13 \, \text{cm} \) (hypotenuse)
  - \( AC = 5 \, \text{cm} \) (leg)
  - Right angle at \( C \)

#### 2. Use the Pythagorean theorem in \( \triangle ACD \):
Since \( \triangle ACD \) is a right triangle, we can apply the Pythagorean theorem:
\[
AD^2 = AC^2 + CD^2
\]
\[
13^2 = 5^2 + CD^2
\]
\[
169 = 25 + CD^2
\]
\[
CD^2 = 144
\]
\[
CD = 12 \, \text{cm}
\]

#### 3. Now, apply the Pythagorean theorem in \( \triangle ABC \):
Using the fact that \( AB = x \), \( AC = 5 \), and \( BC = BD - CD = 19 - 12 = 7 \), we apply the Pythagorean theorem again:
\[
AB^2 = AC^2 + BC^2
\]
\[
x^2 = 5^2 + 7^2
\]
\[
x^2 = 25 + 49
\]
\[
x^2 = 74
\]
\[
x = \sqrt{74} \approx 8.6 \, \text{cm}
\]

### Answer:
The length of \( x \) is approximately **8.6 cm**.

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Would you like further details or clarification on any steps?

Here are 5 related questions to explore this concept further:
1. How does the Pythagorean theorem apply to non-right triangles?
2. What are the conditions for two triangles to be similar?
3. How would the problem change if the angle at \( B \) was not a right angle?
4. Can you solve for the angles of triangle \( ABC \) given this information?
5. How does this problem change if the length of \( CD \) is unknown?

**Tip:** Always double-check your triangle side lengths to ensure you're applying the Pythagorean theorem correctly!

Find the length of x in the given triangle, where AC = 5 cm, AD = 13 cm, AB = x, BD = 19 cm, and the triangle contains two right angles at A and C.

This solution involves finding the length of x in a right triangle using the Pythagorean theorem. The triangle includes known side lengths and requires applying the theorem to solve for the unknown length. Perfect for students in Grades 8-10 learning geometry and the Pythagorean theorem.

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