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The picture shows two similar triangles. Find the unknown measure indicated by a variable. Answer to the nearest tenth. Assume a=2 and b=6.

To solve this problem, we will use the concept of similar triangles. The two triangles are similar, meaning that their corresponding sides are proportional.

Given:
- \( a = 2 \)
- \( b = 6 \)
- The base of the larger triangle (total horizontal side) is \( 5 + a = 5 + 2 = 7 \).
- We need to find \( x \), the height of the smaller triangle.

### Step-by-step solution:

1. **Set up the proportion**: Since the triangles are similar, the ratios of their corresponding sides are equal. Let's write the proportion using the height \( x \) of the smaller triangle and the height \( b = 6 \) of the larger triangle, and the horizontal sides of both triangles:
   \[
   \frac{x}{6} = \frac{5}{7}
   \]

2. **Solve for \( x \)**:
   \[
   x = 6 \times \frac{5}{7}
   \]
   \[
   x = \frac{30}{7} \approx 4.3
   \]

Thus, the height \( x \) of the smaller triangle is approximately **4.3** (to the nearest tenth).

Would you like more details or clarifications?

### Related questions:
1. What is the ratio of the areas of the two triangles?
2. How would the problem change if \( a \) were 3 instead of 2?
3. Can you find the perimeter of the larger triangle?
4. How do you calculate the hypotenuse of both triangles?
5. What is the scale factor between the two triangles?

### Tip:
Always ensure that you identify corresponding sides when working with similar triangles, as this allows for proper proportions to be set up for solving unknowns.

Learn how to solve for the unknown height x in two similar triangles using proportions. This problem involves setting up and solving a ratio for the corresponding sides, given that a=2 and b=6. Suitable for students in Grades 7-9.

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