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

Calculate the area of the trapezoid.

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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