Given the initial triangles, let's solve for sides \( a \) and \( b \) using the Pythagorean theorem and the ratio of similarity.

### Problem Recap:
- **First Triangle:** Sides are \( 9 \), \( 12 \), and \( 15 \) (the hypotenuse).
- **Second Triangle:** One side is given as \( 6 \) and the hypotenuse as \( 10 \).

### Step 1: Establish the Similarity Ratio
From the first triangle:
- Hypotenuse is 15.
- Side corresponding to \( 6 \) in the second triangle is \( 9 \).

The ratio of similarity:
\[
\text{Ratio} = \frac{6}{9} = \frac{2}{3}
\]

### Step 2: Solve for \( a \)
To find \( a \) (the side in the second triangle corresponding to \( 12 \) in the first triangle):
\[
a = \frac{2}{3} \times 12 = 8
\]

### Step 3: Solve for \( b \)
Since we now know:
- \( a = 8 \)
- Hypotenuse \( c = 10 \)

Use the Pythagorean theorem for the second triangle:
\[
c^2 = a^2 + b^2
\]
\[
10^2 = 8^2 + b^2
\]
\[
100 = 64 + b^2
\]
\[
b^2 = 36 \quad \Rightarrow \quad b = 6
\]

### Summary:
- \( a = 8 \)
- \( b = 6 \)

These match the sides of the smaller right triangle.

Would you like more details or have any questions?

### Follow-Up Questions:
1. How does the ratio affect the proportionality of all sides in similar triangles?
2. If you knew only one side and the hypotenuse, how would you find the third side?
3. What is the geometric significance of similar triangles in coordinate geometry?
4. How can these calculations be extended to 3D problems?
5. What are some applications of the Pythagorean theorem in everyday life?

### Tip:
When working with similar triangles, maintaining consistent ratios for all corresponding sides is crucial for accurate results.

Learn how to solve for sides a and b in similar triangles using the Pythagorean theorem and the ratio of similarity. This problem involves calculating the sides of triangles based on given dimensions and similarity ratios. Suitable for Grades 7-9.

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