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

Question

Look at the trapezoid ABCD below.

Length of side AB = a

Side DC is 3 cm longer than AB.

Height (h) = $\frac{1}{2}$ cm

Calculate the length of side AB, given that the area of the trapezoid is 2a cm².

Video Solution

Solution Steps

00:00 Find side AB

00:03 We'll use the formula for calculating trapezoid area

00:07 (Sum of bases(AB+DC) multiplied by height (AD)) divided by 2

00:10 We'll substitute appropriate values and solve for A

00:15 DC size according to the given data

00:29 Half divided by 2 becomes a quarter

00:35 We'll isolate A

00:42 And this is the solution to the question

Step-by-Step Solution

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).

Answer

$\frac{1}{2}$

Related Subjects

Area of a trapezoid Area Trapezoids Types of Trapezoids Areas of Polygons for 7th Grade Area of a right-angled trapezoid Area of an isosceles trapezoid Geometric shapes How do we calculate the area of complex shapes?

Question Types

Area of a Trapezoid: Using ratios for calculation Area of a Trapezoid: Using variables