Trapezoid Area Problem: Find Side Length 'a' Given Area = 2a cm²

To solve this problem, we'll find the length of side AB given the area of the trapezoid. Follow these steps:

• Step 1: Set up the area formula for a trapezoid:
  The area $A$ of a trapezoid is given by the formula $A = \frac{1}{2} \times (\text{Base}_1 + \text{Base}_2) \times \text{Height}$.
• Step 2: Substitute the given information:
  Here, $\text{Base}_1 = AB = a$ and $\text{Base}_2 = DC = a + 3$ cm. The height $h = \frac{1}{2}$ cm. The area is given as $A = 2a$ cm².
• Step 3: Substitute into the formula:
  $2a = \frac{1}{2} \times (a + (a + 3)) \times \frac{1}{2}$
• Step 4: Simplify and solve for $a$:
  $2a = \frac{1}{2} \times (2a + 3) \times \frac{1}{2}$ $2a = \frac{(2a + 3)}{4}$
  Multiply through by 4 to clear the fraction: $8a = 2a + 3$
  Subtract $2a$ from both sides: $6a = 3$
  Divide both sides by 6: $a = \frac{3}{6} = \frac{1}{2}$

Therefore, the length of side AB is $\frac{1}{2}$ cm, and the correct choice is (3).