Similar Quadrilaterals: Finding Original Area When Scaled Area is 12

Q: Why do I need to square the side ratio for areas?

Area is a 2-dimensional measurement, so it scales by the square of linear dimensions. If each side is scaled by factor k, then area is scaled by $ k^2 $. Think of a square: doubling each side makes area 4 times larger!

Q: How do I know which quadrilateral is the original and which is scaled?

Look at the side lengths shown. The quadrilateral with longer sides (10 units) is the original, and the one with shorter sides (4 units) is the scaled version. The scaled version has the smaller area.

Q: What if I get confused about which area goes in the numerator?

Set up your ratio to match the side ratio: $ \frac{\text{smaller area}}{\text{larger area}} = \left(\frac{\text{smaller side}}{\text{larger side}}\right)^2 $. This keeps everything consistent!

Q: Can I work backwards from area to find missing side lengths?

Yes! If you know the area ratio, take its square root to find the side ratio. For example, if area ratio is 0.25, then side ratio is $ \sqrt{0.25} = 0.5 $.

Q: What if the similarity ratio is given as a fraction like 2/5?

Square the entire fraction: $ \left(\frac{2}{5}\right)^2 = \frac{4}{25} $. Then use this as your area ratio. Keep it as a fraction for easier calculations!

Area Scaling with Similarity Ratios

$ABCD∼A^{\prime}B^{\prime}C^{\prime}D^{\prime}$

Given: $S_{A^{\prime}B^{\prime}C^{\prime}D^{\prime}}=12$

Find: $S_{\text{ABCD}}=?$

❤️ Continue Your Math Journey!

We have hundreds of course questions with personalized recommendations + Account 100% premium

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 data

00:08 The ratio of areas equals the square of the similarity ratio

00:22 Let's substitute appropriate values and solve to find the area of polygon 2

00:35 Let's multiply by the reciprocal to isolate the area

00:48 Make sure to square both the numerator and denominator

00:59 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

$ABCD∼A^{\prime}B^{\prime}C^{\prime}D^{\prime}$

Given: $S_{A^{\prime}B^{\prime}C^{\prime}D^{\prime}}=12$

Find: $S_{\text{ABCD}}=?$

Step-by-step solution

To solve this problem, we'll follow these steps:

Step 1: Verify the similarity condition and note given areas and similarity ratio.
Step 2: Calculate the similarity ratio of the sides between the two polygons.
Step 3: Determine the area ratio using the square of the linear ratio.
Step 4: Use the area ratio to find the area of polygon $ABCD$ .

Now, let's work through each step:
Step 1: We know $S_{A'B'C'D'} = 12$ and the similarity ratio as $\frac{4}{10} = 0.4$ .
Step 2: The ratio of corresponding sides is $\frac{A'B'}{AB} = \frac{4}{10} = 0.4$ .
Step 3: The area ratio is the square of the side ratio: $(0.4)^2 = 0.16$ .
Step 4: Let $S_{ABCD}$ be the area of polygon $ABCD$ . Then we have $\frac{12}{S_{ABCD}} = 0.16$ . Solving for $S_{ABCD}$ , we get:

$\begin{aligned} S_{ABCD} &= \frac{12}{0.16} \\ &= 12 \times \frac{100}{16} \\ &= 75. \end{aligned}$

Therefore, the solution to the problem is $S_{ABCD} = 75$ .

Final Answer

Key Points to Remember

Essential concepts to master this topic

Similarity Rule: Area ratio equals square of corresponding side ratio
Technique: If side ratio is $\frac{4}{10} = 0.4$ , then area ratio is $(0.4)^2 = 0.16$
Check: Verify $\frac{12}{75} = 0.16$ matches $\left(\frac{4}{10}\right)^2 = 0.16$ ✓

Common Mistakes

Avoid these frequent errors

Using linear ratio instead of squared ratio for areas
Don't use the side ratio 4:10 directly for areas = wrong answer of 30! Areas scale by the square of the linear ratio, not the linear ratio itself. Always square the side ratio when finding area relationships.

Practice Quiz

Test your knowledge with interactive questions

FAQ

Everything you need to know about this question

Why do I need to square the side ratio for areas?

Area is a 2-dimensional measurement, so it scales by the square of linear dimensions. If each side is scaled by factor k, then area is scaled by $k^2$ . Think of a square: doubling each side makes area 4 times larger!

How do I know which quadrilateral is the original and which is scaled?

Look at the side lengths shown. The quadrilateral with longer sides (10 units) is the original, and the one with shorter sides (4 units) is the scaled version. The scaled version has the smaller area.

What if I get confused about which area goes in the numerator?

Set up your ratio to match the side ratio: $\frac{\text{smaller area}}{\text{larger area}} = \left(\frac{\text{smaller side}}{\text{larger side}}\right)^2$ . This keeps everything consistent!

Can I work backwards from area to find missing side lengths?

Yes! If you know the area ratio, take its square root to find the side ratio. For example, if area ratio is 0.25, then side ratio is $\sqrt{0.25} = 0.5$ .

What if the similarity ratio is given as a fraction like 2/5?

Square the entire fraction: $\left(\frac{2}{5}\right)^2 = \frac{4}{25}$ . Then use this as your area ratio. Keep it as a fraction for easier calculations!

🌟 Unlock Your Math Potential

Get unlimited access to all 18 Mathematics questions, detailed video solutions, and personalized progress tracking.

📹

Unlimited Video Solutions

Step-by-step explanations for every problem

📊

Progress Analytics

Track your mastery across all topics

🚫

Ad-Free Learning

Focus on math without distractions

No credit card required • Cancel anytime