To solve for $x$ based on the given diagram of a triangle, I will first analyze the image:

What we know:

• The triangle has two heights provided: one measuring 6 units and the other 2 units.
• The base length is labeled as $12$ units.
• One of the side lengths is labeled as $2x - 2$.

Step-by-Step Solution:

This is a right triangle, and we can apply the properties of similar triangles because of the two smaller triangles formed within it (due to the height segments).

Step 1: Apply similarity

Since the two triangles formed are similar, the corresponding sides should be proportional.

Let's label the triangles:

• The smaller triangle has a height of 2 and a base of $2x - 2$.
• The larger triangle has a height of 6 and a base of $12$.

We can set up a proportion based on these corresponding sides.

$\frac{2}{6} = \frac{2x - 2}{12}$

Step 2: Solve the proportion

Simplifying the fraction on the left side:

$\frac{1}{3} = \frac{2x - 2}{12}$

Now, cross-multiply to solve for $x$:

$1 \cdot 12 = 3 \cdot (2x - 2)$

$12 = 6x - 6$

Step 3: Isolate $x$

Add 6 to both sides:

$18 = 6x$

Now, divide by 6:

$x = 3$

Conclusion:

The value of $x$ is $\boxed{3}$.

Let me know if you'd like more details or have any questions!

Here are 5 related questions:

1. What are the properties of similar triangles?
2. How do you determine corresponding sides in a proportion?
3. Why is cross-multiplication valid in solving proportions?
4. Can this problem be solved using the Pythagorean theorem?
5. How do we verify that the triangles are similar in this problem?

Tip: Whenever you encounter a geometric problem, look for similarity or congruence between triangles. These concepts often help simplify complex problems!