The image shows two similar right triangles, with a shared height $x$. You are tasked with finding $x$, given that $a = 2$, $b = 6$, and the base of the larger triangle is 5 units.

Step-by-Step Solution:

Step 1: Identify ratios from similar triangles

Because the two triangles are similar, their corresponding sides are proportional. This gives us the relationship between the sides:

$\frac{x}{b} = \frac{5}{5 + a}$

Step 2: Substitute known values

We know that $a = 2$ and $b = 6$, and the full base of the larger triangle is $5 + a = 5 + 2 = 7$. So, substitute these values into the proportion:

$\frac{x}{6} = \frac{5}{7}$

Step 3: Solve for $x$

Now, solve for $x$ by cross-multiplying:

$x \cdot 7 = 5 \cdot 6$

$7x = 30$

$x = \frac{30}{7} \approx 4.3$

Final Answer:

The value of $x$ is approximately 4.3 units (rounded to the nearest tenth).

Would you like more details or have any other questions?

Here are some related questions to extend the problem:

1. How can you verify the triangles are similar?
2. What is the perimeter of the larger triangle?
3. Can you calculate the area of the larger triangle?
4. How would the problem change if $a$ or $b$ were different values?
5. What is the ratio of the areas of the two triangles?

Tip: In problems with similar triangles, always look for corresponding sides to set up ratios.