To address the concept of similar figures and their areas, here are some key points that you might want to include in a quick check or review guide:
Similar Figures:
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Definition: Similar figures are shapes that have the same shape but may have different sizes. Their corresponding angles are equal, and their corresponding sides are in proportion.
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Scale Factor: The ratio of the lengths of corresponding sides of two similar figures is called the scale factor. If figure A has a side length of \( a \) and figure B has a corresponding side length of \( b \), then the scale factor (k) is given by: \[ k = \frac{b}{a} \]
Area Comparison:
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Area of Similar Figures: If two figures are similar and the scale factor between their corresponding lengths is \( k \), then the ratio of their areas will be the square of the scale factor. \[ \text{Area Ratio} = k^2 \]
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Example:
- If one triangle has a side length of 4 cm and a similar triangle has a side length of 8 cm, the scale factor \( k \) is: \[ k = \frac{8}{4} = 2 \]
- The ratio of their areas will be: \[ \text{Area Ratio} = 2^2 = 4 \]
- This means that the area of the larger triangle is 4 times the area of the smaller triangle.
Quick Check Questions:
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True or False: Two shapes that have the same area are always similar figures.
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Multiple Choice: If two similar rectangles have a scale factor of 3, what is the ratio of their areas? A) 3
B) 6
C) 9
D) 12 -
Problem Solving: A square has a side length of 5 cm. A similar square has a side length of 10 cm. What is the ratio of their areas?
Answers:
- False. (They can have the same area but different shapes.)
- C) 9 (Because the area ratio is the square of the scale factor: \( 3^2 = 9 \))
- The area of the first square is \( 5^2 = 25 \) cm², and the area of the second square is \( 10^2 = 100 \) cm². Thus, the ratio of their areas is \( \frac{100}{25} = 4 \).
Feel free to adjust the format or examples based on your needs!