Similarity Of Triangles Worksheet: Practice Problems & Solutions
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Triangles are fundamental shapes in geometry, and understanding their properties is essential for both academic success and real-world applications. A similarity of triangles worksheet is an excellent way to practice and reinforce the concept of triangle similarity, which states that two triangles are similar if their corresponding angles are equal and the lengths of their corresponding sides are proportional. In this article, we’ll explore various practice problems and provide solutions to enhance your understanding of triangle similarity. 🌟
What is Triangle Similarity? 📐
Before delving into the practice problems, let’s clarify what triangle similarity entails. Two triangles are considered similar if:
- Angle-Angle (AA) Criterion: If two angles of one triangle are equal to two angles of another triangle, then the triangles are similar.
- Side-Side-Side (SSS) Criterion: If the ratios of the lengths of the corresponding sides of two triangles are equal, then the triangles are similar.
- Side-Angle-Side (SAS) Criterion: If one angle of a triangle is equal to one angle of another triangle, and the sides including these angles are in proportion, then the triangles are similar.
Importance of Triangle Similarity ✨
Understanding triangle similarity is crucial for several reasons:
- Problem-Solving: It aids in solving real-world problems involving indirect measurement.
- Geometry Foundations: It builds a strong foundation for understanding more complex geometric concepts.
- Applications: It has applications in fields such as architecture, engineering, and various sciences.
Practice Problems on Triangle Similarity 📊
Now that we have a foundational understanding, let’s jump into some practice problems. Here is a table of problems regarding triangle similarity:
<table> <tr> <th>Problem Number</th> <th>Problem Statement</th> </tr> <tr> <td>1</td> <td>Triangle ABC is similar to Triangle DEF. If AB = 5 cm, AC = 8 cm, and DE = 10 cm, find DF.</td> </tr> <tr> <td>2</td> <td>In two similar triangles, the sides of triangle XYZ are 6 cm, 9 cm, and 12 cm. If the longest side of triangle PQR is 18 cm, find the lengths of the other two sides of triangle PQR.</td> </tr> <tr> <td>3</td> <td>Triangle MNO is similar to triangle PQR. If angle M = 50°, angle N = 70°, and angle P = 50°, find angle R.</td> </tr> <tr> <td>4</td> <td>Triangles JKL and MNO are similar. If JK = 4, KL = 6, and MN = 8, find the length of NO.</td> </tr> </table>
Solutions to Practice Problems 🔍
Let’s provide solutions to the above practice problems.
Problem 1 Solution
Given that triangle ABC is similar to triangle DEF, the ratio of the sides is equal.
[ \frac{AB}{DE} = \frac{AC}{DF} ]
Substituting the known values:
[ \frac{5}{10} = \frac{8}{DF} ]
This simplifies to:
[ \frac{1}{2} = \frac{8}{DF} ]
Cross multiplying gives us:
[ DF = 16 \text{ cm} ]
Problem 2 Solution
Triangle XYZ has sides of 6 cm, 9 cm, and 12 cm, which means the scale factor to triangle PQR (with the longest side 18 cm) is:
[ \frac{18}{12} = \frac{3}{2} ]
Now to find the other two sides of triangle PQR:
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For side 6 cm: [ 6 \times \frac{3}{2} = 9 \text{ cm} ]
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For side 9 cm: [ 9 \times \frac{3}{2} = 13.5 \text{ cm} ]
Thus, the sides of triangle PQR are 9 cm and 13.5 cm.
Problem 3 Solution
In triangle MNO and PQR, since M = 50° and P = 50°, we can find angle R using the fact that the angles in a triangle sum to 180°.
Since triangle PQR is similar, we have:
[ \text{Angle R} = 180° - 50° - 70° = 60° ]
Problem 4 Solution
To find side NO in triangle MNO that is similar to JKL, we use the ratio of the sides.
Since JK = 4, KL = 6, and MN = 8, we set up the proportion:
[ \frac{JK}{MN} = \frac{KL}{NO} ]
Substituting the known values:
[ \frac{4}{8} = \frac{6}{NO} ]
This simplifies to:
[ \frac{1}{2} = \frac{6}{NO} ]
Cross multiplying gives us:
[ NO = 12 ]
Key Takeaways ✅
- Triangle similarity is foundational in geometry and has real-world applications.
- Understanding the criteria for triangle similarity is crucial for problem-solving.
- Regular practice with problems and solutions will enhance your understanding and skills.
Incorporating the above elements into your study routine will significantly boost your comprehension of triangle similarity. Happy studying! 🎉