Trapezoid Base Calculation: Finding Bases Given Area 30 cm² and Height 5cm

Q: Why do we use variables like x for the unknown bases?

Using variables helps us set up equations when we have proportional relationships. Since AB is half of DC, we can write AB = x and DC = 2x, making one equation to solve!

Q: What if I mixed up which base is larger?

Always read the problem carefully! 'AB is half of DC' means DC is the longer base. The diagram also shows DC (bottom) is longer than AB (top).

Q: How do I remember the trapezoid area formula?

Think: average of the two bases times height. $ \frac{b_1 + b_2}{2} $ is the average base length, then multiply by height!

Trapezoid Area Formula with Proportional Bases

Given the trapezoid ABCD whose area is equal to 30 cm².

Side AB is equal to half of side DC

The height of the trapezoid is equal to 5cm

How much are the trapeze bases worth?

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Step-by-step video solution

Watch the teacher solve the problem with clear explanations

00:00 Find the bases of trapezoid AB and DC

00:04 Use the formula for calculating trapezoid area

00:08 (Sum of bases(AB+DC) times height(H)) divided by 2

00:12 Substitute appropriate values and solve for DC

00:17 We substituted AB with its size relative to DC according to the given data

00:20 Multiply by 2 to eliminate the fraction

00:25 Isolate DC

00:36 Multiply by the reciprocal to get DC

00:42 This is the size of DC

00:45 Now substitute this size in the sides ratio to find AB

00:52 And this is the solution to the question

Step-by-step written solution

Follow each step carefully to understand the complete solution

Understand the problem

Given the trapezoid ABCD whose area is equal to 30 cm².

Side AB is equal to half of side DC

The height of the trapezoid is equal to 5cm

How much are the trapeze bases worth?

Step-by-step solution

To solve for the bases of the trapezoid, follow these steps:

Let the length of $AB$ be $x$ cm, and since $AB$ is half of $DC$ , let $DC$ be $2x$ cm.

Using the formula for the area of a trapezoid:

$\text{Area} = \frac{1}{2} \times (AB + DC) \times \text{height}$
Substitute the known values: $\frac{1}{2} \times (x + 2x) \times 5 = 30$
Simplify and solve for $x$ : $\frac{1}{2} \times 3x \times 5 = 30$ $7.5x = 30$ $x = \frac{30}{7.5} = 4$
Thus, $AB = x = 4 \, \text{cm}$ and $DC = 2x = 8 \, \text{cm}$ .

Therefore, the lengths of the trapezoid's bases are $AB = 4 \, \text{cm}$ and $DC = 8 \, \text{cm}$ .

Final Answer

DC = 8 , AB=4

Key Points to Remember

Essential concepts to master this topic

Formula: Area = $\frac{1}{2} \times (b_1 + b_2) \times h$ for trapezoid
Substitution: Let AB = x, then DC = 2x gives $\frac{1}{2} \times (x + 2x) \times 5 = 30$
Check: Verify AB = 4, DC = 8: $\frac{1}{2} \times (4 + 8) \times 5 = 30$ ✓

Common Mistakes

Avoid these frequent errors

Using wrong proportional relationship
Don't assume AB equals DC or make DC equal to AB/2 = wrong base sizes! This gives incorrect area calculations. Always read carefully: 'AB is half of DC' means AB = DC/2, so DC = 2×AB.

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 use variables like x for the unknown bases?

Using variables helps us set up equations when we have proportional relationships. Since AB is half of DC, we can write AB = x and DC = 2x, making one equation to solve!

What if I mixed up which base is larger?

Always read the problem carefully! 'AB is half of DC' means DC is the longer base. The diagram also shows DC (bottom) is longer than AB (top).

Can I solve this without using variables?

You could try guess-and-check, but using variables is much more efficient and reliable. It guarantees you'll find the exact answer through systematic algebra.

How do I remember the trapezoid area formula?

Think: average of the two bases times height. $\frac{b_1 + b_2}{2}$ is the average base length, then multiply by height!

What if my calculation gives a decimal answer?

Check your arithmetic! For this problem, the bases should be whole numbers: AB = 4 cm and DC = 8 cm. If you get decimals, review your setup and calculations.

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