In this image, we see two right triangles that are similar, indicated by the fact that they share the same angle and their corresponding sides are proportional.

Given:

• $a = 2$
• $b = 6$
• The length of the larger triangle's base is $5 + a = 5 + 2 = 7$

We need to find $x$, which represents the height of the smaller triangle (the red segment).

Step 1: Set up a proportion

Since the two triangles are similar, the ratios of their corresponding sides are equal. The ratio between the sides of the smaller triangle and the larger triangle can be set up as: $\frac{x}{b} = \frac{a}{5 + a}$

Substitute the known values of $a = 2$ and $b = 6$: $\frac{x}{6} = \frac{2}{7}$

Step 2: Solve for $x$

To solve for $x$, multiply both sides of the equation by 6: $x = 6 \times \frac{2}{7} = \frac{12}{7} \approx 1.71$

Answer:

The value of $x$ is approximately 1.7 (to the nearest tenth).

Let me know if you need any more details or further explanation!

Here are 5 related questions for deeper understanding:

1. What other methods can be used to confirm the similarity of triangles?
2. How does the Pythagorean Theorem apply to right triangles in this problem?
3. If $a$ or $b$ changed, how would that affect $x$?
4. Can we generalize a formula for $x$ based on any values of $a$ and $b$?
5. How could this setup be applied in real-world measurement problems?

Tip: When working with similar triangles, always check the proportionality of corresponding sides.