Rectangle Area Problem: Finding EBFD Area Using 1/7 Ratio and Parallel Lines

Q: Why isn't the area ratio 1/49 if EB is 1/7 of AB?

Great question! When we scale both dimensions of a shape, the area ratio becomes the square of the dimension ratio. But here, only the width changes while the height stays the same, so area ratio equals dimension ratio: $ \frac{1}{7} $.

Q: How do I know which dimension is changing?

Look at the given information carefully! Since $ EB = \frac{1}{7}AB $ and EF is parallel to BD, point E divides the top side in the ratio 1:7, affecting only the width of rectangle EBFD.

Q: What does EF || BD mean for the problem?

The parallel lines symbol (||) tells us that EF is parallel to BD. This creates a smaller rectangle EBFD that has the same height as the original rectangle ABCD, but a different width.

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

Think logically: the smaller rectangle should have less area than the original. Since 7 $ 7 \times 7 = 49 $ ✓

Rectangle Area Ratios with Parallel Lines

The area of the rectangle below is equal to 49.

$EB=\frac{1}{7}AB$

$EF\Vert BD$

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

00:00 Determine the area of the rectangle EBFD

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

00:11 Substitute in the relevant values according to the given data and solve to find the area

00:19 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 49.

$EB=\frac{1}{7}AB$

$EF\Vert BD$

Calculate the area of the rectangle EBFD.

Step-by-step solution

Since AB is 7 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 $\frac{1}{7}$

$S_{\text{ABCD}}=7\times S_{EBFD}$

Let's input the known data into the formula:

$49=7\times S_{\text{EBFD}}$

$S_{\text{EBFD}}=7$

Final Answer

Key Points to Remember

Essential concepts to master this topic

Ratio Rule: When dimensions scale by ratio, area scales by same ratio
Technique: If EB = (1/7)AB, then area EBFD = (1/7) × 49
Check: Verify 7 × 7 = 49 confirms smaller rectangle area ✓

Common Mistakes

Avoid these frequent errors

Thinking area ratio equals dimension ratio squared
Don't calculate area as $(\frac{1}{7})^2 \times 49 = 1$ ! This treats it like scaling both dimensions, but only one dimension changes here. Always use the same ratio for area when only one dimension scales proportionally.

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 1/49 if EB is 1/7 of AB?

Great question! When we scale both dimensions of a shape, the area ratio becomes the square of the dimension ratio. But here, only the width changes while the height stays the same, so area ratio equals dimension ratio: $\frac{1}{7}$ .

How do I know which dimension is changing?

Look at the given information carefully! Since $EB = \frac{1}{7}AB$ and EF is parallel to BD, point E divides the top side in the ratio 1:7, affecting only the width of rectangle EBFD.

What does EF || BD mean for the problem?

The parallel lines symbol (||) tells us that EF is parallel to BD. This creates a smaller rectangle EBFD that has the same height as the original rectangle ABCD, but a different width.

Can I solve this without using ratios?

You could find actual dimensions first, but ratios are much faster! If you know the areas are proportional to the dimension ratios, you can directly calculate: $\frac{1}{7} \times 49 = 7$ .

How do I check if my answer makes sense?

Think logically: the smaller rectangle should have less area than the original. Since 7 < 49, our answer makes sense. Also verify: $7 \times 7 = 49$ ✓

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