Calculate Rectangle EBFD Area: Using 1/6 Proportion and Parallel Lines

Q: Why isn't the area ratio 1/36 if the side ratio is 1/6?

Great question! You might be thinking of similar rectangles where area ratios equal the square of side ratios. But here, we have proportional rectangles with the same height - only the width changes by the ratio 1:6.

Q: How do I know that EF is parallel to BD affects the area?

The parallel condition $ EF \parallel BD $ tells us that EBFD forms a rectangle, not just any quadrilateral. This means we can directly apply the proportion rule to find its area.

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

Then the area of rectangle EBFD would be $ \frac{1}{3} \times 36 = 12 $. The key is that area ratios equal side ratios when rectangles share the same height.

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

You could, but it's much easier to use proportions directly! If you know the original area is 36 and the width ratio is 1:6, you immediately get $ \frac{36}{6} = 6 $.

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

Ask yourself: Should the smaller rectangle have a smaller area? Since EB is only $ \frac{1}{6} $ of AB, the area 6 should be less than 36. ✓

Rectangle Area Proportions with Parallel Lines

The area of a rectangle is 36.

$EB=\frac{1}{6}AB$

$EF\Vert BD$

Calculate the area of 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 areas between the rectangles equals the ratio of the sides

00:12 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 a rectangle is 36.

$EB=\frac{1}{6}AB$

$EF\Vert BD$

Calculate the area of rectangle EBFD.

Step-by-step solution

Since AB is 6 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 $\frac{1}{6}$

$S_{\text{ABCD}}=6\times S_{EBFD}$

Let's input the known data into the formula:

$36=6\times S_{\text{EBFD}}$

$S_{\text{EBFD}}=6$

Final Answer

Key Points to Remember

Essential concepts to master this topic

Proportion Rule: When sides are in ratio 1:6, areas are also in ratio 1:6
Technique: If EB = $\frac{1}{6}AB$ , then Area(EBFD) = $\frac{1}{6} \times 36 = 6$
Check: Verify that 6 × 6 = 36 for the original rectangle ✓

Common Mistakes

Avoid these frequent errors

Confusing side ratios with area ratios
Don't think that if EB = $\frac{1}{6}AB$ , then the area is $\frac{1}{36}$ of the original = wrong ratio! This confuses linear measurements with area measurements. Always remember that when corresponding sides are in ratio 1:6, the areas are also in the same ratio 1:6.

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 1/36 if the side ratio is 1/6?

Great question! You might be thinking of similar rectangles where area ratios equal the square of side ratios. But here, we have proportional rectangles with the same height - only the width changes by the ratio 1:6.

How do I know that EF is parallel to BD affects the area?

The parallel condition $EF \parallel BD$ tells us that EBFD forms a rectangle, not just any quadrilateral. This means we can directly apply the proportion rule to find its area.

What if EB was 1/3 of AB instead?

Then the area of rectangle EBFD would be $\frac{1}{3} \times 36 = 12$ . The key is that area ratios equal side ratios when rectangles share the same height.

Can I solve this by finding actual dimensions first?

You could, but it's much easier to use proportions directly! If you know the original area is 36 and the width ratio is 1:6, you immediately get $\frac{36}{6} = 6$ .

How do I check if my answer makes sense?

Ask yourself: Should the smaller rectangle have a smaller area? Since EB is only $\frac{1}{6}$ of AB, the area 6 should be less than 36. ✓

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