This image shows a diagram of two right triangles. We are tasked with solving for $x$, which represents the length of segment $DE$ in the smaller triangle.

Breakdown:

• Larger right triangle $\triangle ABC$:

  • $AB = 9$
  • $BC = x + 3$ (since $BE = x$ and $EC = 3$)
  • $\angle ABC = 90^\circ$

• Smaller right triangle $\triangle BDE$:

  • $BD = 6$
  • $DE = x$
  • $\angle BDE = 90^\circ$

Step-by-step solution:

1. Use the Pythagorean Theorem in the smaller triangle $\triangle BDE$: $BD^2 = BE^2 + DE^2$ Substituting the values $BD = 6$ and $DE = x$: $6^2 = 3^2 + x^2$ $36 = 9 + x^2$ $x^2 = 27$ $x = \sqrt{27} = 3\sqrt{3}$

Thus, $x = 3\sqrt{3}$ is the exact value of $x$. Approximating, this is about $x \approx 5.2$.

Would you like a deeper explanation or have any specific questions about the steps?

Here are 5 related questions:

1. How is the Pythagorean theorem applied in right triangles?
2. What is the importance of the right angle in solving for $x$ in this problem?
3. How would the solution change if the side lengths were different?
4. What if $x$ were located in a non-right triangle? Could we still use the Pythagorean theorem?
5. Can you explain how to simplify $\sqrt{27}$ to $3\sqrt{3}$?

Tip: Whenever possible, leave the answer in exact form (like $3\sqrt{3}$) unless an approximation is explicitly asked for!