Rectangle Area Problem: Finding EBFD When EB=1/4AB and Area=16

Q: Why isn't the area ratio (1/4)² = 1/16?

Great question! The area ratio is not (1/4)² because only one dimension changes. EB = 1/4 AB, but EF equals the full height of the rectangle. So area ratio = 1/4, not 1/16.

Q: How do I know EF equals the full height?

The problem states $ EF\parallel BD $, meaning EF is parallel to BD. Since ABCD is a rectangle, EF must span the entire height from top to bottom.

Q: What does 'parallel lines' mean in this context?

When $ EF\parallel BD $, it means EF runs in the same direction as BD (vertically). This tells us that EF connects the top and bottom sides of the rectangle, giving it the full height.

Q: Can I solve this without finding the actual dimensions?

Yes! You don't need to find length and width separately. Since the area ratio equals the side ratio (1/4), just calculate: $ \frac{1}{4} \times 16 = 4 $

Q: How do I verify my answer is correct?

Check that rectangle EBFD area + rectangle AEFC area = 16. If EBFD = 4, then AEFC should equal 12, and 4 + 12 = 16 ✓

Rectangle Area Ratios with Parallel Lines

The area of the rectangle below is: 16.

$EB=\frac{1}{4}AB$

$EF\Vert BD$

Calculate the area of the rectangle EBFD.

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Step-by-step video solution

Watch the teacher solve the problem with clear explanations

00:11 Let's find the area of rectangle E B F D.

00:15 The areas of the rectangles are proportional to the lengths of their sides.

00:22 Now, substitute the given values, and solve to find the area.

00:30 Great job! That's how we find the area.

Step-by-step written solution

Follow each step carefully to understand the complete solution

Understand the problem

The area of the rectangle below is: 16.

$EB=\frac{1}{4}AB$

$EF\Vert BD$

Calculate the area of the rectangle EBFD.

Step-by-step solution

Since AB is 4 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 and the larger one is $\frac{1}{4}$

$S_{\text{ABCD}}=4\times S_{EBFD}$

Let's input the known data into the formula:

$16=4\times S_{\text{EBFD}}$

$S_{\text{EBFD}}=4$

Final Answer

Key Points to Remember

Essential concepts to master this topic

Ratio Rule: When EB = 1/4 AB, area ratio equals side ratio
Technique: If side ratio is 1/4, then area EBFD = 1/4 × 16 = 4
Check: Verify EB × EF = 4 matches calculated area ✓

Common Mistakes

Avoid these frequent errors

Calculating area using both dimensions incorrectly
Don't think the area ratio is (1/4)² = 1/16 because both dimensions change! Only one dimension (EB) is 1/4 of AB, while EF equals the full height. Always identify which dimensions actually change in the problem.

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

Why isn't the area ratio (1/4)² = 1/16?

Great question! The area ratio is not (1/4)² because only one dimension changes. EB = 1/4 AB, but EF equals the full height of the rectangle. So area ratio = 1/4, not 1/16.

How do I know EF equals the full height?

The problem states $EF\parallel BD$ , meaning EF is parallel to BD. Since ABCD is a rectangle, EF must span the entire height from top to bottom.

What does 'parallel lines' mean in this context?

When $EF\parallel BD$ , it means EF runs in the same direction as BD (vertically). This tells us that EF connects the top and bottom sides of the rectangle, giving it the full height.

Can I solve this without finding the actual dimensions?

Yes! You don't need to find length and width separately. Since the area ratio equals the side ratio (1/4), just calculate: $\frac{1}{4} \times 16 = 4$

How do I verify my answer is correct?

Check that rectangle EBFD area + rectangle AEFC area = 16. If EBFD = 4, then AEFC should equal 12, and 4 + 12 = 16 ✓

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