Rectangle Area Problem: Finding Smaller Section When EB=1/3AB and EF∥BD

Q: How do I know which rectangle is the smaller one?

Look at the given fraction! Since $ EB = \frac{1}{3}AB $, point E divides AB so that EB is the shorter segment. Rectangle EBDF uses this shorter width, making it smaller.

Q: Why does the area ratio equal the length ratio?

Because both rectangles have the same height! When EF ∥ BD, the height stays constant. So if the width is $ \frac{1}{3} $ as long, the area is also $ \frac{1}{3} $ as big.

Q: What does the parallel symbol ∥ tell me?

EF ∥ BD means EF is parallel to BD. This creates two rectangles with the same height, so we can use simple proportions to find their areas.

Q: Can I solve this without using fractions?

Yes! Think of it as: large rectangle = 3 × small rectangle. So 60 = 3 × small rectangle, which gives small rectangle = 20.

Q: What if EB was 2/3 of AB instead?

Then rectangle EBDF would be larger! Its area would be $ \frac{2}{3} \times 60 = 40 $, and rectangle AEFC would be the smaller one with area 20.

Rectangle Areas with Parallel Line Proportions

The area of the rectangle below is equal to 60.

$EB=\frac{1}{3}AB$

$EF\Vert BD$

Calculate the area of the smaller rectangle.

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Step-by-step video solution

Watch the teacher solve the problem with clear explanations

00:00 Determine the area of the rectangle EBFD

00:03 The ratio of the areas between the rectangles equals the ratio of sides

00:13 Substitute in the relevant values according to the given data and solve for the area

00:20 This is the solution

Step-by-step written solution

Follow each step carefully to understand the complete solution

Understand the problem

The area of the rectangle below is equal to 60.

$EB=\frac{1}{3}AB$

$EF\Vert BD$

Calculate the area of the smaller rectangle.

Step-by-step solution

Since AB is 3 times larger than EB, the area of rectangle EBDF will be smaller than the area of rectangle ABCD accordingly

In other words, the ratio between the smaller rectangle to the larger one is $\frac{1}{3}$

$S_{\text{ABCD}}=3\times S_{EBFD}$

Let's input the known data into the formula:

$60=3\times S_{\text{EBFD}}$

$S_{\text{EBFD}}=20$

Final Answer

Key Points to Remember

Essential concepts to master this topic

Proportion Rule: When $EB = \frac{1}{3}AB$ , areas have same ratio
Technique: If large area = 60, then small area = $\frac{1}{3} \times 60 = 20$
Check: Verify that 3 × 20 = 60 for the total area ✓

Common Mistakes

Avoid these frequent errors

Confusing which rectangle is smaller
Don't assume rectangle AEFC is the smaller one = wrong answer of 40! Since EB = 1/3 of AB, rectangle EBDF takes up only 1/3 of the width, making it smaller. Always identify which dimension is being divided by the fraction.

Practice Quiz

Test your knowledge with interactive questions

Look at the rectangle ABCD below.

Side AB is 6 cm long and side BC is 4 cm long.

What is the area of the rectangle?

FAQ

Everything you need to know about this question

How do I know which rectangle is the smaller one?

Look at the given fraction! Since $EB = \frac{1}{3}AB$ , point E divides AB so that EB is the shorter segment. Rectangle EBDF uses this shorter width, making it smaller.

Why does the area ratio equal the length ratio?

Because both rectangles have the same height! When EF ∥ BD, the height stays constant. So if the width is $\frac{1}{3}$ as long, the area is also $\frac{1}{3}$ as big.

What does the parallel symbol ∥ tell me?

EF ∥ BD means EF is parallel to BD. This creates two rectangles with the same height, so we can use simple proportions to find their areas.

Can I solve this without using fractions?

Yes! Think of it as: large rectangle = 3 × small rectangle. So 60 = 3 × small rectangle, which gives small rectangle = 20.

What if EB was 2/3 of AB instead?

Then rectangle EBDF would be larger! Its area would be $\frac{2}{3} \times 60 = 40$ , and rectangle AEFC would be the smaller one with area 20.

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