Rectangle Area Problem: Finding EBFD When EB = 1/5 AB and Area = 50

Q: Why is the area ratio the same as the length ratio?

Since rectangle EBFD has the same height as ABCD but $ \frac{1}{5} $ the width, its area is exactly $ \frac{1}{5} $ of the original area. The height cancels out in the ratio!

Q: How do I know EF is parallel to BD?

The notation $ EF \parallel BD $ tells us EF is parallel to BD. This means EBFD forms a proper rectangle with the same height as the original rectangle ABCD.

Q: What if EB was 1/3 of AB instead?

Then the area of EBFD would be $ \frac{1}{3} \times 50 = \frac{50}{3} $ or about 16.67. The area ratio always equals the width ratio when heights are the same.

Q: Can I solve this by finding the actual dimensions?

You could, but it's much harder! You'd need to solve $ length \times width = 50 $ with infinitely many solutions. The ratio method gives you the answer directly.

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

Ask yourself: Should EBFD be smaller than ABCD? Yes, because EB is only 1/5 of AB. Since 10

Area Ratios with Proportional Segments

The area of a rectangle is 50.

$EB=\frac{1}{5}AB$

$EF\Vert BD$

Work out the area of the rectangle EBFD.

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Step-by-step video solution

Watch the teacher solve the problem with clear explanations

00:10 Let's find the area of rectangle E B F D.

00:13 Remember, the area ratios of rectangles are the same as the ratios of their side lengths.

00:20 Now, substitute the given values for the sides into the formula, and solve to get the area.

00:31 And that's how you calculate the area of this rectangle!

Step-by-step written solution

Follow each step carefully to understand the complete solution

Understand the problem

The area of a rectangle is 50.

$EB=\frac{1}{5}AB$

$EF\Vert BD$

Work out the area of the rectangle EBFD.

Step-by-step solution

Since AB is 5 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 $\frac{1}{5}$

$S_{\text{ABCD}}=5\times S_{EBFD}$

Let's input the known data into the formula:

$50=5\times S_{\text{EBFD}}$

$S_{\text{EBFD}}=10$

Final Answer

Key Points to Remember

Essential concepts to master this topic

Proportion Rule: When EB = 1/5 AB, rectangle EBFD has 1/5 the area
Ratio Method: Area ABCD = 5 × Area EBFD, so 50 = 5 × Area EBFD
Check: Verify 10 × 5 = 50, confirming area of EBFD is 10 ✓

Common Mistakes

Avoid these frequent errors

Calculating area using individual dimensions instead of ratios
Don't try to find actual length and width values like AB = 10, AD = 5 = wrong approach! This makes the problem unnecessarily complex and often leads to errors. Always use the direct ratio relationship: if EB = 1/5 AB, then Area EBFD = 1/5 × Area ABCD.

Practice Quiz

Test your knowledge with interactive questions

Look at the rectangle below.

Side AB is 2 cm long and side BC has a length of 7 cm.

What is the perimeter of the rectangle?

FAQ

Everything you need to know about this question

Why is the area ratio the same as the length ratio?

Since rectangle EBFD has the same height as ABCD but $\frac{1}{5}$ the width, its area is exactly $\frac{1}{5}$ of the original area. The height cancels out in the ratio!

How do I know EF is parallel to BD?

The notation $EF \parallel BD$ tells us EF is parallel to BD. This means EBFD forms a proper rectangle with the same height as the original rectangle ABCD.

What if EB was 1/3 of AB instead?

Then the area of EBFD would be $\frac{1}{3} \times 50 = \frac{50}{3}$ or about 16.67. The area ratio always equals the width ratio when heights are the same.

Can I solve this by finding the actual dimensions?

You could, but it's much harder! You'd need to solve $length \times width = 50$ with infinitely many solutions. The ratio method gives you the answer directly.

How do I check if my answer makes sense?

Ask yourself: Should EBFD be smaller than ABCD? Yes, because EB is only 1/5 of AB. Since 10 < 50, your answer passes the sense check! ✓

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