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The problem you uploaded shows a geometric figure composed of a rectangle divided into two trapezoids and a triangle, with the goal of finding the value of \(x\).

### Given Information:
- The rectangle is split into two trapezoids and a triangle.
- The heights of the trapezoids and triangle are not directly given but can be inferred.
- The areas of the trapezoids are \(135 \, \text{cm}^2\) each.
- The area of the triangle is \(90 \, \text{cm}^2\).
- The top lengths of the trapezoids are \(12 \, \text{cm}\) each.
- The bottom lengths of the trapezoids are \(x \, \text{cm}\) each.

### Step-by-Step Solution:

#### Step 1: Calculate the height of the trapezoids
The area of a trapezoid is given by:
\[
\text{Area} = \frac{1}{2} \times (\text{Base}_1 + \text{Base}_2) \times \text{Height}
\]
For the given trapezoid:
\[
135 = \frac{1}{2} \times (12 + x) \times h
\]
This simplifies to:
\[
270 = (12 + x) \times h
\]
\[
h = \frac{270}{12 + x}
\]

#### Step 2: Calculate the height of the triangle
The area of a triangle is given by:
\[
\text{Area} = \frac{1}{2} \times \text{Base} \times \text{Height}
\]
For the triangle:
\[
90 = \frac{1}{2} \times (x + x) \times h_{\text{triangle}}
\]
This simplifies to:
\[
90 = x \times h_{\text{triangle}}
\]
\[
h_{\text{triangle}} = \frac{90}{x}
\]

Since the trapezoids and the triangle share the same height \( h \), we equate the height expressions from the two figures:
\[
\frac{270}{12 + x} = \frac{90}{x}
\]

#### Step 3: Solve for \(x\)
Cross-multiply to eliminate the fractions:
\[
270x = 90(12 + x)
\]
\[
270x = 1080 + 90x
\]
\[
180x = 1080
\]
\[
x = \frac{1080}{180} = 6 \, \text{cm}
\]

### Final Answer:
\[
x = 6 \, \text{cm}
\]

Would you like more details or have any other questions? Here are some related questions to expand your understanding:

1. How would you calculate the perimeter of the original rectangle?
2. What would happen to the value of \(x\) if the area of the triangle was doubled?
3. Can you find the height of the trapezoid using the value of \(x\)?
4. How would you determine the area of the rectangle if \(x\) were given as 10 cm instead?
5. How can the equation for the height \( h \) of the trapezoid be used to verify the consistency of the solution?

**Tip:** Always double-check your equations for consistency by substituting values back into the original equations. This helps to confirm your solution.

The query submitted by the user: The problem you uploaded shows a geometric figure composed of a rectangle divided into two trapezoids and a triangle, with the goal of finding the value of \(x\).

Learn how to solve a geometric problem involving a rectangle divided into two trapezoids and a triangle to find the value of \(x\). This problem covers area calculations for trapezoids and triangles, suitable for high school students.

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