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In the given image, we see two right triangles sharing a common height, and both triangles seem to be similar because corresponding angles are congruent. The lengths of the bases of the triangles are expressed algebraically: \( 2x + 1 \) and \( 3x - 5 \).

### To solve this:
1. **Identify the triangles:**
   - The two triangles are right-angled.
   - There are two sets of corresponding sides: one on the base and the hypotenuse.
   - The red markings on the hypotenuse indicate that the two hypotenuses of the smaller triangles are equal, confirming the triangles are similar by AA similarity (angle-angle similarity).
   
2. **Use triangle similarity:**
   - Since the triangles are similar, the ratio of corresponding sides should be equal.
   
   \[
   \frac{\text{Base of smaller triangle}}{\text{Base of larger triangle}} = \frac{\text{Height of smaller triangle}}{\text{Height of larger triangle}} = \frac{\text{Hypotenuse of smaller triangle}}{\text{Hypotenuse of larger triangle}}
   \]
   
   From the diagram, the bases are \( 2x + 1 \) (smaller triangle) and \( 3x - 5 \) (larger triangle).

   Therefore, the ratio of the sides becomes:
   \[
   \frac{2x + 1}{3x - 5}
   \]
   
   If the heights and hypotenuses are proportional by the same factor, you could set up a proportionality equation to solve for \(x\).

Would you like to proceed with solving for \( x \) or need further clarification on the relationships between the triangles?

Solve for x in a triangle problem where two similar right triangles share a common height, with base lengths of 2x + 1 and 3x - 5.

Explore how to solve for x in a problem involving similar right triangles with base lengths of 2x + 1 and 3x - 5. This problem demonstrates the use of triangle similarity and proportional reasoning, suitable for students in Grades 9-11.

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