Similar Figures Practice Worksheet Answers Explained
Table of Contents :
When it comes to mastering geometric concepts, understanding similar figures is paramount. Similar figures, such as triangles, rectangles, and other polygons, share the same shape but differ in size. In this article, we will explore the concept of similar figures, provide a detailed explanation of practice worksheet answers, and explain the calculations involved. Let's dive in!
Understanding Similar Figures 📐
Definition of Similar Figures
Similar figures are shapes that have the same form but are scaled versions of each other. This means that the corresponding angles are equal, and the lengths of corresponding sides are in proportion. For example, if triangle ABC is similar to triangle DEF, it can be expressed as:
- ∠A = ∠D
- ∠B = ∠E
- ∠C = ∠F
Key Properties of Similar Figures
- Corresponding Angles: All corresponding angles are equal.
- Proportional Sides: The ratios of the lengths of corresponding sides are equal.
The Importance of Similar Figures 🎯
Understanding similar figures is crucial in various fields, including engineering, architecture, and art. It allows for accurate scaling and design, ensuring that proportions are maintained in smaller or larger versions of a figure.
Similar Figures Practice Worksheet
Example Problems 📝
Here we present a practice worksheet that features common scenarios involving similar figures. We will discuss the answers and the reasoning behind them.
Problem 1: Triangle ABC is similar to Triangle DEF. If AB = 4 cm, AC = 5 cm, and DE = 8 cm, find the length of DF.
Solution:
Since the triangles are similar, we can set up the proportion based on the lengths of their corresponding sides:
[ \frac{AB}{DE} = \frac{AC}{DF} ]
Substituting the known values:
[ \frac{4}{8} = \frac{5}{DF} ]
Cross-multiply to solve for DF:
[ 4 \cdot DF = 5 \cdot 8 \ 4 \cdot DF = 40 \ DF = 10 \text{ cm} ]
Answer: DF = 10 cm
Problem 2: Two rectangles, Rectangle A and Rectangle B, are similar. The length of Rectangle A is 12 cm, and the length of Rectangle B is 18 cm. If the width of Rectangle A is 8 cm, find the width of Rectangle B.
Solution:
First, set up the proportion based on the lengths:
[ \frac{Length\ of\ A}{Length\ of\ B} = \frac{Width\ of\ A}{Width\ of\ B} ]
Substituting the values we know:
[ \frac{12}{18} = \frac{8}{Width\ of\ B} ]
Cross-multiply:
[ 12 \cdot Width\ of\ B = 8 \cdot 18 \ 12 \cdot Width\ of\ B = 144 \ Width\ of\ B = 12 \text{ cm} ]
Answer: Width of Rectangle B = 12 cm
Practice Worksheet Table
To further clarify the relationships between corresponding sides, here is a summary table showcasing these relationships for different geometric shapes.
<table> <tr> <th>Shape</th> <th>Corresponding Sides Ratio</th> <th>Corresponding Angles</th> <th>Example</th> </tr> <tr> <td>Triangle</td> <td>AB/DE = AC/DF</td> <td>∠A = ∠D</td> <td>ABC ~ DEF</td> </tr> <tr> <td>Rectangle</td> <td>LengthA/LengthB = WidthA/WidthB</td> <td>All angles = 90°</td> <td>Rectangle A ~ Rectangle B</td> </tr> <tr> <td>Square</td> <td>SideA/SideB</td> <td>All angles = 90°</td> <td>Square A ~ Square B</td> </tr> </table>
Important Note: Understanding the relationships among corresponding sides and angles is crucial for solving similar figures problems. Make sure to carefully note the relationships before proceeding with calculations. 💡
Advanced Practice: Word Problems 🔍
Problem 3: A model car is 1:20 scale of a real car. If the real car is 5 m long, how long is the model car?
Solution:
To find the model car’s length, use the ratio:
[ \frac{Model\ Length}{Real\ Length} = \frac{1}{20} ]
Let (x) be the model length:
[ \frac{x}{5} = \frac{1}{20} ]
Cross-multiply:
[ 20x = 5 \ x = \frac{5}{20} = 0.25 \text{ m or } 25 \text{ cm} ]
Answer: The model car is 25 cm long.
Conclusion
By practicing problems related to similar figures, students can enhance their geometric reasoning skills. Familiarity with proportions and angle relationships not only aids in solving mathematical problems but also prepares learners for practical applications in real-world scenarios. Remember, similar figures can be found everywhere, and mastering this concept is crucial for any budding mathematician or scientist! Keep practicing, and soon you’ll be able to tackle similar figures with confidence! 🎓✨