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.