To solve for $x$ using proportions, let's analyze the given right triangles in the diagram.

Step 1: Identify the triangles

We have two right triangles in the diagram:

1. The smaller right triangle on the left, with sides 5, 10, and 7.
2. The larger right triangle which includes the entire base $x$.

Step 2: Set up the proportion

Since the two triangles are similar (they share an angle and both have a right angle), the corresponding sides are proportional.

We can set up the proportion using the corresponding sides: $\frac{\text{side opposite the shared angle in small triangle}}{\text{side adjacent to the shared angle in small triangle}} = \frac{\text{side opposite the shared angle in large triangle}}{\text{side adjacent to the shared angle in large triangle}}$

Using the lengths provided: $\frac{10}{5} = \frac{(10 + 7)}{x}$

Step 3: Solve for $x$

Simplify and solve the proportion: $\frac{10}{5} = \frac{17}{x}$ $2 = \frac{17}{x}$ Multiply both sides by $x$ to get: $2x = 17$ Finally, divide by 2: $x = \frac{17}{2} = 8.5$

Final Answer

The value of $x$ is $8.5$.

Would you like further details or have any additional questions? Here are some related questions you might be interested in:

1. How do you prove two triangles are similar using the AA (Angle-Angle) criterion?
2. What is the significance of the Pythagorean theorem in right triangles?
3. Can you apply the same method to find the length of other sides in similar triangles?
4. How do you determine the scale factor between two similar triangles?
5. What are some real-world applications of similar triangles and proportions?
6. How do you use trigonometric ratios in right triangles?
7. How does the concept of similarity extend to non-right triangles?
8. What are the properties of a right triangle that make solving proportions easier?

Tip: When solving proportions, always cross-multiply to simplify your calculations and verify the correctness of your solution.