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The problem you uploaded involves two similar right triangles. The goal is to find the unknown height \( x \) given that:

- \( a = 4 \)
- \( b = 5 \)
- The bottom segment of the larger triangle is split into two parts: \( 5 \) (the base of the smaller triangle) and \( a = 4 \) (the remainder of the base of the larger triangle).

Since the two triangles are similar, the sides are proportional. We can set up the following proportion using corresponding sides of the two triangles:

\[
\frac{x}{b} = \frac{5}{5 + a}
\]

Substitute \( b = 5 \) and \( a = 4 \):

\[
\frac{x}{5} = \frac{5}{5 + 4}
\]
\[
\frac{x}{5} = \frac{5}{9}
\]

Now solve for \( x \) by multiplying both sides of the equation by 5:

\[
x = 5 \times \frac{5}{9} = \frac{25}{9} \approx 2.8
\]

So, \( x \approx 2.8 \).

Would you like further details on the reasoning or another example? Here are some related questions to explore further:

1. What are the conditions for two triangles to be similar?
2. How do you find missing sides in similar triangles using proportions?
3. Can the Pythagorean theorem also be used in this problem?
4. How would the result change if \( a \) or \( b \) were different?
5. What if the triangles were not right triangles? How would the approach change?

**Tip:** Always double-check your proportions when dealing with similar triangles to ensure you're matching the corresponding sides correctly.

The figure shown below contains two similar triangles. Find the unknown measure indicated by a variable. Answer to the nearest tenth. (Assume a = 4 and b = 5.)

Learn how to find the unknown height of a triangle using proportions with similar triangles. This problem involves setting up proportions between corresponding sides, using values a = 4 and b = 5, and solving for x. Suitable for students in Grades 8-10.

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