Geometry 11.2 Practice B Answers

Geometry 11.2 Practice B Answers: A Deep Dive into Similar Triangles

This comprehensive guide provides detailed solutions and explanations for Geometry 11.2 Practice B problems, focusing on the concept of similar triangles. Understanding similar triangles is crucial for various applications in geometry, trigonometry, and even real-world scenarios like surveying and architecture. We'll break down the key principles, walk through example problems, and explore the underlying mathematical logic. This resource is designed to help you not just find the answers but also deeply grasp the concepts behind them.

Introduction: What are Similar Triangles?

Similar triangles are triangles that have the same shape but not necessarily the same size. This means their corresponding angles are congruent (equal), and their corresponding sides are proportional. The ratio of corresponding sides is called the scale factor. Identifying similar triangles relies on understanding postulates and theorems like AA (Angle-Angle), SAS (Side-Angle-Side), and SSS (Side-Side-Side) similarity.

Key Theorems and Postulates for Similar Triangles

Before diving into the practice problems, let's review the fundamental theorems and postulates:

• AA Similarity Postulate: If two angles of one triangle are congruent to two angles of another triangle, then the triangles are similar.

• SAS Similarity Theorem: If two sides of one triangle are proportional to two sides of another triangle and the included angles are congruent, then the triangles are similar.

• SSS Similarity Theorem: If three sides of one triangle are proportional to three sides of another triangle, then the triangles are similar.

Geometry 11.2 Practice B: Problem Breakdown and Solutions

Since I don't have access to a specific Geometry 11.2 Practice B worksheet, I'll create representative problems that cover the common types of questions found in such exercises. Remember to always refer to your specific worksheet for the exact problems and their corresponding numbers.

Problem 1: Using AA Similarity

Problem: In the figure, ΔABC and ΔDEF are given with ∠A ≅ ∠D and ∠B ≅ ∠E. If AB = 6, BC = 8, and DE = 9, find EF.

Solution:

1. Identify Similar Triangles: Since ∠A ≅ ∠D and ∠B ≅ ∠E, by the AA Similarity Postulate, ΔABC ~ ΔDEF.

2. Set up Proportions: Because the triangles are similar, the ratios of their corresponding sides are equal. We can set up the proportion: AB/DE = BC/EF.

3. Substitute and Solve: Plugging in the given values, we get 6/9 = 8/EF. Cross-multiplying gives 6EF = 72. Therefore, EF = 12.

Problem 2: Using SAS Similarity

Problem: In ΔABC and ΔXYZ, AB = 10, BC = 15, XY = 6, YZ = 9, and ∠B ≅ ∠Y. Are the triangles similar?

Solution:

1. Check for Proportional Sides: We compare the ratios of corresponding sides: AB/XY = 10/6 = 5/3 and BC/YZ = 15/9 = 5/3. The ratios are equal.

2. Check for Congruent Included Angles: We are given that ∠B ≅ ∠Y.

3. Conclusion: Since two sides are proportional and the included angles are congruent, by the SAS Similarity Theorem, ΔABC ~ ΔXYZ.

Problem 3: Using SSS Similarity

Problem: The sides of ΔPQR are PQ = 12, QR = 18, and RP = 15. The sides of ΔSTU are ST = 8, TU = 12, and US = 10. Are the triangles similar?

Solution:

1. Check for Proportional Sides: We check the ratios of corresponding sides: PQ/ST = 12/8 = 3/2; QR/TU = 18/12 = 3/2; RP/US = 15/10 = 3/2. All ratios are equal.

2. Conclusion: Since all three sides are proportional, by the SSS Similarity Theorem, ΔPQR ~ ΔSTU.

Problem 4: Finding Missing Side Lengths using Similar Triangles

Problem: Two triangles, ΔABC and ΔADE, are similar. If AB = 4, BC = 6, AC = 8, and AD = 6, find DE and AE.

Solution:

1. Set up Proportions: Since ΔABC ~ ΔADE, we can set up proportions using corresponding sides: AB/AD = BC/DE = AC/AE.

2. Solve for DE: We use the proportion AB/AD = BC/DE: 4/6 = 6/DE. Cross-multiplying gives 4DE = 36, so DE = 9.

3. Solve for AE: We use the proportion AB/AD = AC/AE: 4/6 = 8/AE. Cross-multiplying gives 4AE = 48, so AE = 12.

Problem 5: Real-World Application of Similar Triangles

Problem: A tree casts a shadow 20 feet long. At the same time, a 6-foot tall person casts a shadow 4 feet long. How tall is the tree?

Solution:

1. Set up Similar Triangles: The tree and its shadow form a right-angled triangle, similar to the triangle formed by the person and their shadow.

2. Set up Proportion: Let 'h' be the height of the tree. We can set up the proportion: h/20 = 6/4.

3. Solve for h: Cross-multiplying gives 4h = 120, so h = 30 feet. The tree is 30 feet tall.

Explanation of Scientific Principles

The concept of similar triangles is based on Euclidean geometry, specifically on the properties of angles and the ratios of sides in triangles. The postulates and theorems discussed above are foundational and are derived from Euclid's axioms. The principle of proportionality is central to understanding similarity; it states that the ratio between corresponding sides remains constant for similar figures. This principle has far-reaching applications in various branches of mathematics and science.

Frequently Asked Questions (FAQ)

• Q: What's the difference between congruent and similar triangles?

  • A: Congruent triangles are identical in both shape and size (all corresponding sides and angles are equal). Similar triangles have the same shape but may differ in size (corresponding angles are equal, and corresponding sides are proportional).

• Q: Can any two triangles be similar?

  • A: No. Two triangles must satisfy at least one of the similarity postulates (AA, SAS, SSS) to be considered similar.

• Q: How do I know which sides are corresponding in similar triangles?

  • A: Corresponding sides are the sides opposite congruent angles. If you've established similarity using the AA postulate, you'll need to examine the correspondence of angles to determine the corresponding sides.

• Q: Is it possible to have similar triangles without knowing all the angles?

  • A: Yes, the SAS and SSS similarity theorems allow you to prove similarity without knowing all the angles.

Conclusion: Mastering Similar Triangles

Understanding similar triangles is a cornerstone of geometry. By mastering the AA, SAS, and SSS similarity postulates and theorems, and by practicing solving problems that involve finding missing side lengths and determining similarity, you build a strong foundation for more advanced geometric concepts. Remember to always clearly identify corresponding sides and angles, correctly set up proportions, and carefully solve for unknown variables. Consistent practice is key to developing proficiency in solving problems related to similar triangles. This understanding extends beyond the classroom and has practical applications in diverse fields, making it a valuable skill to acquire.