To solve for the missing variable $x$ in this triangle:

Step 1: Recognize the geometry

This is a right triangle. The lengths of the two legs (both 6 units) and the hypotenuse $H$ form the triangle.

Step 2: Use the Pythagorean theorem

The Pythagorean theorem states: $H^2 = \text{leg}_1^2 + \text{leg}_2^2$

Substitute the given values: $H^2 = 6^2 + 6^2$ $H^2 = 36 + 36 = 72$

Solve for $H$: $H = \sqrt{72} = 6\sqrt{2} \, \text{(approximately } 8.49\text{)}.$

Step 3: Solve for $x$

The segment $x$ corresponds to the horizontal leg of the triangle. Since the smaller triangle is similar to the original, the proportion of sides is preserved.

Using the smaller triangle:

• The smaller hypotenuse is 6.
• Corresponding hypotenuse for the original is $H = 6\sqrt{2}$.

Set up the proportion for side $x$: $\frac{x}{6} = \frac{6}{6\sqrt{2}}$

Simplify: $x = \frac{6 \cdot 6}{6\sqrt{2}} = \frac{36}{\sqrt{2}}$

Rationalize the denominator: $x = \frac{36\sqrt{2}}{2} = 18\sqrt{2}$

So: $x \approx 25.46$

Let me know if you'd like further clarification!

Related Questions:

1. What is the value of $H$ expressed as a simplified radical?
2. How do you confirm that triangles are similar in this problem?
3. Why do we use rationalization in this context?
4. What is the geometric interpretation of $x$?
5. How does the Pythagorean theorem work for other right triangles?

Tip:

Always check triangle similarity when solving problems involving proportional parts or scaling.