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

Q: Why do we have two bases in a trapezoid?

A trapezoid has two parallel sides called bases. In this problem, AB = a and DC = a + 3. The area formula uses both bases because trapezoids aren't rectangles - they have different lengths on top and bottom!

Q: Why is the height so small compared to the bases?

The height of $ \frac{1}{2} $ cm might seem tiny, but it's just the perpendicular distance between the parallel sides. Trapezoids can be wide but not very tall!

Q: How do I check if my answer makes sense?

Substitute a = $ \frac{1}{2} $ back into the area formula. You should get: $ \frac{1}{4} \times (\frac{1}{2} + 3.5) = 1 $ cm², and 2a = $ 2 \times \frac{1}{2} = 1 $ cm² ✓

Q: Can trapezoids have negative side lengths?

Never! Side lengths must always be positive. If you get a negative answer, check your algebra - you probably made a sign error or calculation mistake somewhere.

Step-by-step video solution

Watch the teacher solve the problem with clear explanations

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 written solution

Follow each step carefully to understand the complete solution

1

Understand the problem

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.

2

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).

3

Final Answer

$\frac{1}{2}$

Key Points to Remember

Essential concepts to master this topic

Formula: Trapezoid area equals one-half times sum of bases times height
Technique: Substitute known values: $2a = \frac{1}{2} \times (a + (a + 3)) \times \frac{1}{2}$
Check: Verify AB = $\frac{1}{2}$ gives area = $2 \times \frac{1}{2} = 1$ cm² ✓

Common Mistakes

Avoid these frequent errors

Forgetting to multiply the area formula by both fractions
Don't multiply by just one $\frac{1}{2}$ = wrong area calculation! Students often forget that height $\frac{1}{2}$ combines with the formula's $\frac{1}{2}$ to give $\frac{1}{4}$ . Always multiply all parts of the formula by the given height value.

Practice Quiz

Test your knowledge with interactive questions

Given the following trapezoid:

Calculate the area of the trapezoid ABCD.

FAQ

Everything you need to know about this question

Why do we have two bases in a trapezoid?

+

A trapezoid has two parallel sides called bases. In this problem, AB = a and DC = a + 3. The area formula uses both bases because trapezoids aren't rectangles - they have different lengths on top and bottom!

How do I handle the algebra when the area contains the variable?

+

Set up your equation carefully! When area = 2a and your formula gives an expression with a, you get an equation to solve. Don't get confused - treat 2a as just another number to work with.

What does it mean when DC is 3 cm longer than AB?

+

This means DC = AB + 3. Since AB = a, we write DC = a + 3. This is a relationship between the two bases that helps us write everything in terms of the single variable a.

Why is the height so small compared to the bases?

+

The height of $\frac{1}{2}$ cm might seem tiny, but it's just the perpendicular distance between the parallel sides. Trapezoids can be wide but not very tall!

How do I check if my answer makes sense?

+

Substitute a = $\frac{1}{2}$ back into the area formula. You should get: $\frac{1}{4} \times (\frac{1}{2} + 3.5) = 1$ cm², and 2a = $2 \times \frac{1}{2} = 1$ cm² ✓

Can trapezoids have negative side lengths?

+

Never! Side lengths must always be positive. If you get a negative answer, check your algebra - you probably made a sign error or calculation mistake somewhere.

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

Trapezoid Area Formula with Algebraic Expressions

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Step-by-step video solution

Step-by-step written solution

Understand the problem

Step-by-step solution

Final Answer

Key Points to Remember

Common Mistakes

Practice Quiz

FAQ

Why do we have two bases in a trapezoid?

How do I handle the algebra when the area contains the variable?

What does it mean when DC is 3 cm longer than AB?

Why is the height so small compared to the bases?

How do I check if my answer makes sense?

Can trapezoids have negative side lengths?

🌟 Unlock Your Math Potential

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