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?

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.