Similar Figures: Calculate Area MPNO Given ABCD Area=1848

Q: Why do I need to square the scale factor for area?

Areas are two-dimensional, so they scale by the square of the linear scale factor. If sides are in ratio 1:2, areas are in ratio $ 1^2:2^2 = 1:4 $.

Q: How do I identify corresponding sides in similar figures?

Look for matching positions in the figures. In this problem, DE corresponds to OQ because they're both marked segments in similar positions within their respective figures.

Q: What if I get the scale factor upside down?

If your final area is larger than the given area, you probably flipped the ratio. The smaller figure should have the smaller area - check which figure the unknown area belongs to!

Q: Can I use any corresponding sides to find the scale factor?

Yes! All corresponding sides in similar figures have the same ratio. Pick the sides that are clearly labeled or easiest to work with.

Q: How do I set up the proportion correctly?

Set up as: $ \frac{\text{Area of larger figure}}{\text{Area of smaller figure}} = \frac{\text{(larger side)}^2}{\text{(smaller side)}^2} $. Keep the same figure in numerator on both sides!

Similar Figures with Scale Factor Calculation

$ONMP∼\text{ABCD}$

DE = 7

OQ = 2.8

If the area of ABCD is 1848, then what is the area MPNO?

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Step-by-step video solution

Watch the teacher solve the problem with clear explanations

00:00 Find the area of MPNO

00:03 Similar polygons according to given data

00:10 The ratio of areas equals the square of the similarity ratio

00:22 Substitute appropriate values and solve to find the area

00:32 Multiply by the reciprocal to isolate the area

00:40 Square both numerator and denominator

00:45 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

$ONMP∼\text{ABCD}$

DE = 7

OQ = 2.8

If the area of ABCD is 1848, then what is the area MPNO?

Step-by-step solution

We are given that the triangles are similar, so we can compare the ratio between DE and OQ:

2.8:7

Raised to the 2nd power

$7.84:49$

We are given that the area of ABCD is 1848. Now we calculate the area of MPNO:

$\frac{1848}{S}=\frac{49}{7.84}$

$S49=1848\times7.84$

$S49=14488.32$

$S=295.68$

Final Answer

295.68

Key Points to Remember

Essential concepts to master this topic

Scale Factor Rule: Find ratio of corresponding sides first
Area Scaling: Square the scale factor: 2.8:7 becomes 7.84:49
Verification: Check that 1848 × 7.84 ÷ 49 = 295.68 ✓

Common Mistakes

Avoid these frequent errors

Using linear scale factor for area ratios
Don't use the side ratio 2.8:7 directly for areas = wrong answer of 739.2! Areas scale by the square of the linear ratio, not the linear ratio itself. Always square the scale factor when finding area ratios.

Practice Quiz

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FAQ

Everything you need to know about this question

Why do I need to square the scale factor for area?

Areas are two-dimensional, so they scale by the square of the linear scale factor. If sides are in ratio 1:2, areas are in ratio $1^2:2^2 = 1:4$ .

How do I identify corresponding sides in similar figures?

Look for matching positions in the figures. In this problem, DE corresponds to OQ because they're both marked segments in similar positions within their respective figures.

What if I get the scale factor upside down?

If your final area is larger than the given area, you probably flipped the ratio. The smaller figure should have the smaller area - check which figure the unknown area belongs to!

Can I use any corresponding sides to find the scale factor?

Yes! All corresponding sides in similar figures have the same ratio. Pick the sides that are clearly labeled or easiest to work with.

How do I set up the proportion correctly?

Set up as: $\frac{\text{Area of larger figure}}{\text{Area of smaller figure}} = \frac{\text{(larger side)}^2}{\text{(smaller side)}^2}$ . Keep the same figure in numerator on both sides!

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