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The diagram shows a right triangle with a smaller rectangle within it. Here’s the breakdown of the given values:

- The height of the triangle is \( 2.856 \) units.
- The length of the base of the entire triangle is \( 14.5 + 5.5 = 20 \) units.
- The smaller rectangle has a base of \( 5.5 \) units and a height of \( 3.447 \) units.
- The task is to find the length of \( x \), which is the remaining height of the triangle from the base of the triangle minus the height of the rectangle.

### Steps:
1. The triangle’s height decreases linearly as you move along its base due to its shape.
2. Using similar triangles, we can relate the larger triangle to the smaller part.
3. Set up a proportion to find the unknown height \( x \) using the ratios of corresponding sides.

Let me calculate this for you.The calculated value for \( x \) is negative, which suggests there may be a misunderstanding in how the heights are being interpreted or related in the diagram. Let me reassess the setup:

Since the triangle's height decreases as we move towards the smaller rectangle, and the right triangle's hypotenuse defines this decrease, I will use a proportion between the large triangle and the smaller one to correct the approach.

### Corrected Approach:
- The height of the triangle decreases uniformly, and we can use the base lengths to set up a proportion.
  
Let me correct this by setting up a ratio based on the similar triangles.It seems like I can’t do more advanced data analysis right now. Please try again later. However, I can explain the corrected approach if you'd like to try solving it yourself.

Would you like to go over the steps manually? If so, here’s a quick outline:

1. The total base of the triangle is 20 units, and the base of the smaller rectangle is 5.5 units.
2. The remaining triangle next to the rectangle has a base of \( 20 - 5.5 = 14.5 \) units.
3. Using similar triangles, the proportion would be:

   \[
   \frac{\text{Height of full triangle}}{\text{Base of full triangle}} = \frac{x}{\text{Base of smaller triangle}}
   \]

4. Solve for \( x \).

Let me know if you need more assistance with the steps! Here are some follow-up questions:

1. What concept are similar triangles based on?
2. How do you set up a proportion between two triangles?
3. Can you explain why heights and bases of triangles are proportional?
4. What would happen if the height of the rectangle were larger than the triangle’s height?
5. How does the Pythagorean theorem apply to right triangles like this one?

**Tip:** Always double-check units and values when working with geometric figures and proportions.

Find the length of X using the given dimensions of the triangle and rectangle in the diagram.

Learn how to find the missing length X in a right triangle using proportions and similar triangles. The given dimensions are 2.856 for the total height and 14.5 for part of the base. This problem covers geometric concepts, including proportions and the Similar Triangles Theorem. Suitable for Grades 9-11 students.

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