Practice Quiz 7 Unit 2 Dilations and Similarity Answers
Understanding dilations and similarity is a cornerstone of geometry, especially in Unit 2, where students learn to analyze transformations and relationships between shapes. This practice quiz focuses on key concepts like scale factors, properties of similar figures, and problem-solving techniques. Below, you’ll find explanations of the quiz questions along with detailed answers to reinforce your mastery of these topics And it works..
The official docs gloss over this. That's a mistake Small thing, real impact..
Key Concepts in Dilations and Similarity
Before diving into the quiz, let’s review the foundational ideas:
Dilations
A dilation is a transformation that resizes a figure by a scale factor while preserving its shape. The center of dilation determines how the figure is scaled. If the scale factor is greater than 1, the figure enlarges; if it’s between 0 and 1, it shrinks. Key properties include:
- Proportional sides: All corresponding sides are scaled by the same factor.
- Preserved angles: Angles remain unchanged.
- Orientation: The figure does not flip or rotate unless combined with other transformations.
Similarity
Two figures are similar if:
- All corresponding angles are equal.
- All corresponding sides are proportional.
The scale factor of similarity determines how much one figure is enlarged or reduced compared to the other.
Practice Quiz Questions and Answers
Question 1
A dilation with a scale factor of 2 is applied to a triangle with side lengths 3 cm, 4 cm, and 5 cm. What are the side lengths of the dilated triangle?
Answer:
When a dilation is applied, each side length is multiplied by the scale factor.
- New side lengths = 3 × 2 = 6 cm, 4 × 2 = 8 cm, 5 × 2 = 10 cm.
Final Answer: 6 cm, 8 cm, 10 cm.
Question 2
Triangle ABC is similar to triangle DEF. The ratio of their corresponding sides is 3:5. If AB = 9 cm, what is the length of DE?
Answer:
The ratio of similarity is 3:5 (ABC:DEF).
Using proportions:
[
\frac{AB}{DE} = \frac{3}{5} \implies \frac{9}{DE} = \frac{3}{5}
]
Cross-multiplying:
[
3 \times DE = 9 \times 5 \implies DE = \frac{45}{3} = 15 \text{ cm}.
]
Final Answer: 15 cm Small thing, real impact..
Question 3
A rectangle with length 12 cm and width 8 cm is dilated by a scale factor of 0.5. What are the new dimensions?
Answer:
Multiply both dimensions by the scale factor:
- New length = 12 × 0.5 = 6 cm
- New width = 8 × 0.5 = 4 cm
Final Answer: 6 cm (length) and 4 cm (width).
Question 4
Two triangles are similar, and their areas are 18 cm² and 50 cm². What is the ratio of their corresponding side lengths?
Answer:
The ratio of areas of similar figures is the square of the ratio of their corresponding sides. Let the side ratio be ( x ).
[
x^2 = \frac{18}{50} = \frac{9}{25} \implies x = \sqrt{\frac{9}{25}} = \frac{3}{5}.
]
Final Answer: 3:5 Worth knowing..
Question 5
A point P(4, 6) is dilated with a scale factor of 3 centered at the origin. What are the coordinates of the image point P’?
Answer:
For a dilation centered at the origin, multiply coordinates by the scale factor:
- ( x' = 4 \times 3 = 12 )
- ( y' = 6 \times 3 = 18 )
Final Answer: P’(12, 18).
Question 6
If two polygons are similar with a perimeter ratio of 4:7, what is the ratio of their areas?
Answer:
The ratio of perimeters equals the ratio of corresponding sides. The area ratio is the square of this:
[
(4:7)^2 = 16:49.
]
Final Answer: 16:49 Worth knowing..
Question 7
A square is dilated by a scale factor of -2. Describe the effect on the square.
Answer:
A negative scale factor reflects the figure across the center of dilation and scales it by the absolute value. Here, the square is reflected and enlarged to twice its original size.
Final Answer: The square is reflected and enlarged by a factor of 2 The details matter here. That's the whole idea..
Question 8
In similar triangles, if one angle measures 45° and another measures 60°, what is the measure of the third angle?
Answer:
Similar triangles have equal corresponding angles. The sum of angles in a triangle is 180°:
[
45° + 60° + \text{Third angle} = 180° \implies \text{Third angle} = 75°.
]
Final Answer: 75°.
Question 9
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Question 10
Two similar rectangles have areas of 45 cm² and 80 cm². The length of the corresponding side in the smaller rectangle is 9 cm. What is the length of the corresponding side in the larger rectangle?
Answer:
For similar figures, the ratio of their areas equals the square of the ratio of their corresponding linear dimensions Took long enough..
Let the ratio of the side lengths be ( r = \dfrac{\text{side of larger}}{\text{side of smaller}} ) Small thing, real impact..
[ \frac{\text{Area of larger}}{\text{Area of smaller}} = r^{2} \quad\Longrightarrow\quad \frac{80}{45}=r^{2} ]
Simplify the fraction:
[ \frac{80}{45}= \frac{16}{9} ]
Thus
[ r = \sqrt{\frac{16}{9}} = \frac{4}{3} ]
Now apply this ratio to the known side length:
[ \text{Side of larger} = 9 \text{ cm} \times \frac{4}{3} = 12 \text{ cm} ]
Final Answer: 12 cm.
Conclusion
These problems illustrate the fundamental relationships governing similarity and dilation in geometry. By understanding how scale factors affect lengths, perimeters, and areas—and how negative scale factors introduce reflections—we can solve a wide range of transformation problems efficiently. Mastery of these concepts provides a solid foundation for more advanced topics in Euclidean geometry and coordinate transformations Small thing, real impact..
