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.

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!