We can solve this using the Pythagorean Theorem and some geometry.

The diagram shows two right triangles. Let's label them:

1. The larger right triangle has sides of length $6$ (the height), $x$ (the base), and the hypotenuse is the side of length $5 + 2 = 7$ (total height).

2. The smaller right triangle has legs $2$ (base) and $5$ (height), and its hypotenuse lies along the hypotenuse of the larger triangle.

Step 1: Apply the Pythagorean Theorem to the smaller triangle.

For the smaller triangle: $\text{Hypotenuse}^2 = \text{Base}^2 + \text{Height}^2$ $h^2 = 2^2 + 5^2 = 4 + 25 = 29$ Thus, the hypotenuse of the smaller triangle is $\sqrt{29}$.

Step 2: Use similar triangles.

Because the two triangles are similar (they share the right angle and have corresponding angles due to the height), we can set up a proportion for corresponding sides: $\frac{x}{6} = \frac{2}{\sqrt{29}}$ Solving for $x$: $x = 6 \times \frac{2}{\sqrt{29}} = \frac{12}{\sqrt{29}}$ Now rationalize the denominator: $x = \frac{12}{\sqrt{29}} \times \frac{\sqrt{29}}{\sqrt{29}} = \frac{12\sqrt{29}}{29}$

Thus, $x = \frac{12\sqrt{29}}{29}$.

Would you like further details or have any questions?

Related Questions:

1. How do you find the hypotenuse of a right triangle?
2. What are similar triangles, and how do you use them to solve problems?
3. How is the Pythagorean Theorem applied in right triangles?
4. Why do we rationalize the denominator, and how is it done?
5. Can this method be used to solve any right triangle problems?

Tip: When solving for unknowns in triangles, always check for similarity or congruency first, as these provide useful ratios to set up proportions.