Similar Rectangles: Finding Area Ratio with 9:8 Side Length Proportion

Q: Why do we square the ratio instead of just using 9:8?

Because area is two-dimensional! When you scale a rectangle, both length and width change by the same factor. So if sides are 9:8, then areas are $ 9 \times 9 : 8 \times 8 = 81:64 $.

Q: What does 'multiply the lengths by 9' mean in the question?

This is asking about the final step - after finding that areas are in ratio 81:64, the question asks what happens next. But the answer is still 81:64 because that's the area ratio of the similar rectangles.

Q: How do I remember to square the ratio?

Think about it this way: Length ratios → square for area, cube for volume. Area has 2 dimensions, so you raise the ratio to the 2nd power!

Q: Would this work for any similar shapes?

Yes! For any similar figures, if the side ratio is a:b, then the area ratio is always $ a^2:b^2 $. This works for triangles, circles, polygons - everything!

Q: What if the ratio was given as a fraction like 9/8?

Same process! Square the fraction: $ \left(\frac{9}{8}\right)^2 = \frac{81}{64} $. You can write this as either 81:64 or $ \frac{81}{64} $.

Step-by-step video solution

Watch the teacher solve the problem with clear explanations

00:00 Find the ratio of areas between the large rectangles

00:03 We multiplied the rectangles according to the given data

00:13 The ratio of rectangle areas equals the similarity ratio squared

00:33 Square both numerator and denominator

00:41 And this is the solution to the question

Step-by-step written solution

Follow each step carefully to understand the complete solution

1

Understand the problem

Rectangle ABCD is similar to Rectangle EFGH.

The ratio between the lengths of their sides is 9:8.

If we multiply the lengths of the sides by 9, then what is the ratio between the areas of the large rectangles?

2

Step-by-step solution

We raise to the power of 2:

$9^2:8^2=81:64$

3

Final Answer

81:64

Key Points to Remember

Essential concepts to master this topic

Similarity Rule: Area ratio equals side ratio squared for similar figures
Technique: Square the side ratio: $9^2:8^2 = 81:64$
Check: Area ratio should be larger than side ratio when ratio > 1 ✓

Common Mistakes

Avoid these frequent errors

Using side ratio directly for area ratio
Don't use 9:8 as the area ratio = completely wrong answer! Area involves two dimensions (length × width), so the ratio gets multiplied by itself. Always square the side ratio to get the area ratio.

Practice Quiz

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FAQ

Everything you need to know about this question

Why do we square the ratio instead of just using 9:8?

+

Because area is two-dimensional! When you scale a rectangle, both length and width change by the same factor. So if sides are 9:8, then areas are $9 \times 9 : 8 \times 8 = 81:64$ .

What does 'multiply the lengths by 9' mean in the question?

+

This is asking about the final step - after finding that areas are in ratio 81:64, the question asks what happens next. But the answer is still 81:64 because that's the area ratio of the similar rectangles.

How do I remember to square the ratio?

+

Think about it this way: Length ratios → square for area, cube for volume. Area has 2 dimensions, so you raise the ratio to the 2nd power!

Would this work for any similar shapes?

+

Yes! For any similar figures, if the side ratio is a:b, then the area ratio is always $a^2:b^2$ . This works for triangles, circles, polygons - everything!

What if the ratio was given as a fraction like 9/8?

+

Same process! Square the fraction: $\left(\frac{9}{8}\right)^2 = \frac{81}{64}$ . You can write this as either 81:64 or $\frac{81}{64}$ .

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