Similar Polygons Area Problem: Finding Blue Triangle Area Given Green Area of 64

Q: Why do we square the side ratio to get the area ratio?

Because area is a two-dimensional measurement! When you scale a polygon, both length and width get multiplied by the scale factor. So area gets multiplied by scale factor × scale factor = scale factor².

Q: How do I find the side ratio from the diagram?

Look for corresponding sides - sides in the same position on both polygons. Here, the green polygon has side 12 and the blue polygon has corresponding side 3, giving ratio $ \frac{12}{3} = 4 $.

Q: What if the polygons are positioned differently?

The orientation doesn't matter! Similar polygons have the same shape regardless of position or rotation. Just match up corresponding sides by their relative positions.

Q: Can I use any pair of corresponding sides?

Yes! In similar polygons, all corresponding sides have the same ratio. Pick whichever pair is clearly labeled in your diagram.

Q: How do I remember the area ratio formula?

Think "area = length × width". If both dimensions scale by factor k, then area scales by $ k \times k = k^2 $. Area ratio = (side ratio)²!

Similar Polygons with Area Ratios

What is the area of the blue polygon, given that the two polygons are similar and the area of the green polygon is 64?

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Step-by-step video solution

Watch the teacher solve the problem with clear explanations

00:00 Find the area of the polygon

00:03 The polygons are similar according to the given

00:07 We want to find the similarity ratio

00:11 We'll break down 12 into factors (3) and (4) and reduce

00:14 This is the similarity ratio

00:18 The similarity ratio squared equals the area ratio squared

00:34 We'll substitute appropriate values and solve to find the area of polygon 2

00:39 We'll multiply by the reciprocal to isolate the area

00:53 And this is the solution to the question

Step-by-step written solution

Follow each step carefully to understand the complete solution

Understand the problem

What is the area of the blue polygon, given that the two polygons are similar and the area of the green polygon is 64?

Step-by-step solution

From the similarity, it follows that:

$\frac{12}{3}=4$

Therefore:

$\frac{64}{S}=16$

$\frac{64}{16}=S$

$S=4$

Final Answer

Key Points to Remember

Essential concepts to master this topic

Ratio Rule: Area ratio equals square of corresponding side ratio
Technique: Find side ratio first: $\frac{12}{3} = 4$ , then square for area ratio
Check: Verify $64 \div 16 = 4$ matches calculated area ✓

Common Mistakes

Avoid these frequent errors

Using side ratio directly as area ratio
Don't use side ratio 4:1 directly for areas = wrong answer of 16! Area scales by the square of the side ratio, not the ratio 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 side ratio to get the area ratio?

Because area is a two-dimensional measurement! When you scale a polygon, both length and width get multiplied by the scale factor. So area gets multiplied by scale factor × scale factor = scale factor².

How do I find the side ratio from the diagram?

Look for corresponding sides - sides in the same position on both polygons. Here, the green polygon has side 12 and the blue polygon has corresponding side 3, giving ratio $\frac{12}{3} = 4$ .

What if the polygons are positioned differently?

The orientation doesn't matter! Similar polygons have the same shape regardless of position or rotation. Just match up corresponding sides by their relative positions.

Can I use any pair of corresponding sides?

Yes! In similar polygons, all corresponding sides have the same ratio. Pick whichever pair is clearly labeled in your diagram.

What if I get a decimal answer?

That's fine! Convert to a fraction if needed, or check if the decimal makes sense. Here, getting 4 is a nice whole number, which we can verify: $4 \times 16 = 64$ ✓

How do I remember the area ratio formula?

Think "area = length × width". If both dimensions scale by factor k, then area scales by $k \times k = k^2$ . Area ratio = (side ratio)²!

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