Solve for X: Trapezoid Area of 12 cm² with Triangle Relationship

To solve this problem, we need to use the formula for the area of a trapezoid and the relationships given in the problem.

Step 1: Identify the given information
From the diagram and problem statement, we have:

• Upper base (top of trapezoid) = $2x$
• Lower base (bottom of trapezoid) = $4x$
• Height of trapezoid = $x$
• Area of trapezoid = $12$ square cm

Step 2: Verify the relationships
Let's confirm the stated relationships:

• Lower base is 2 times upper base: $4x = 2 \times 2x = 4x$ ✓
• Lower base is 4 times height: $4x = 4 \times x = 4x$ ✓

Step 3: Apply the trapezoid area formula
The area of a trapezoid is given by:
$A = \frac{1}{2}(b_1 + b_2) \times h$
where $b_1$ and $b_2$ are the two parallel bases and $h$ is the height.

Step 4: Substitute the values
Substituting our expressions into the formula:
$12 = \frac{1}{2}(2x + 4x) \times x$

Step 5: Simplify and solve for x
$12 = \frac{1}{2}(6x) \times x$
$12 = \frac{6x^2}{2}$
$12 = 3x^2$
$x^2 = \frac{12}{3}$
$x^2 = 4$
$x = 2$ (taking the positive root since x represents a length)

Step 6: Verify the solution
When $x = 2$:

• Upper base = $2x = 4$ cm
• Lower base = $4x = 8$ cm
• Height = $x = 2$ cm
• Area = $\frac{1}{2}(4 + 8) \times 2 = \frac{1}{2}(12) \times 2 = 12$ square cm ✓

Therefore, the value of x equals $x = 2$.