https://file.aiquickdraw.com/mathbot/file/17304452518434al7npx4.jpg

To determine the value of \( x \) in this diagram, let's analyze it in steps.

### Step 1: Understand the Configuration
The figure shows two right triangles that share a similar orientation. The smaller right triangle on top has side lengths \( 3 \) (opposite the right angle) and \( 2 \) (adjacent to the right angle). The larger right triangle has side lengths \( x \) (hypotenuse) and \( 6 \) (adjacent to the right angle).

Since both triangles are right triangles and have a common angle, they are similar. Therefore, we can use the ratios of their corresponding sides to find \( x \).

### Step 2: Set Up the Proportion
In similar triangles, the ratios of corresponding sides are equal. Thus, we can write the proportion based on the two triangles:

\[
\frac{\text{opposite side of small triangle}}{\text{adjacent side of small triangle}} = \frac{\text{opposite side of large triangle}}{\text{adjacent side of large triangle}}
\]

Substituting the known values, we get:

\[
\frac{3}{2} = \frac{x}{6}
\]

### Step 3: Solve for \( x \)
Cross-multiply to find \( x \):

\[
3 \cdot 6 = 2 \cdot x
\]
\[
18 = 2x
\]
\[
x = \frac{18}{2} = 9
\]

### Answer
The value of \( x \) is \( 9 \).

Would you like a further breakdown of the steps or have any questions?

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### Further Questions for Practice
1. What would the value of \( x \) be if the sides of the smaller triangle were scaled by a factor of 2?
2. How do you calculate the hypotenuse if the values for the other two sides of a triangle are known?
3. Why are these triangles similar, and what are the criteria for triangle similarity?
4. What would happen to \( x \) if the adjacent side of the larger triangle changed to 8?
5. Can this proportion method be used if one of the triangles is not a right triangle?

### Tip
Remember that similar triangles have corresponding angles equal and their sides in proportion. This property is very useful for finding unknown side lengths.

Learn how to find the value of x in a geometric figure with two similar right triangles. This solution covers key geometric concepts such as triangle similarity, side ratios, and solving for unknown sides using proportions. Suitable for students in Grades 7-9.

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