7 3 Similar Triangles Practice

7.3 Similar Triangles: Mastering the Concepts and Practice Problems

Understanding similar triangles is a cornerstone of geometry, crucial for solving a wide range of problems in mathematics and its applications in fields like architecture, engineering, and surveying. This comprehensive guide will delve into the concept of similar triangles, focusing specifically on practice problems related to the 7.3 section often found in geometry textbooks. We will explore the fundamental theorems, provide step-by-step solutions to various problems, and offer insights to enhance your problem-solving skills. Mastering similar triangles opens doors to more advanced geometric concepts and builds a strong foundation for future mathematical endeavors.

Introduction to Similar Triangles

Two triangles are considered similar if their corresponding angles are congruent (equal in measure) and their corresponding sides are proportional. This means that one triangle is essentially a scaled version of the other. This proportional relationship between corresponding sides is often expressed as a ratio. The notation for similar triangles uses the symbol ~. For example, if triangle ABC is similar to triangle DEF, we write ΔABC ~ ΔDEF.

The key theorems underpinning similar triangles are:

• Angle-Angle (AA) Similarity Postulate: If two angles of one triangle are congruent to two angles of another triangle, then the triangles are similar. Since the sum of angles in a triangle is always 180°, proving two angles are congruent automatically implies the third angle is also congruent.

• Side-Side-Side (SSS) Similarity Theorem: If the ratios of the lengths of the corresponding sides of two triangles are equal, then the triangles are similar.

• Side-Angle-Side (SAS) Similarity Theorem: If the ratio of the lengths of two sides of one triangle is equal to the ratio of the lengths of two corresponding sides of another triangle, and the included angles are congruent, then the triangles are similar.

Step-by-Step Problem Solving Approach

Let's approach similar triangle problems systematically. Here's a five-step method:

1. Identify Similar Triangles: Carefully examine the given diagrams and information to identify pairs of triangles that might be similar. Look for congruent angles (marked with arcs or stated explicitly) or proportional sides.

2. State the Similarity: Once you've identified similar triangles, write down the similarity statement using the correct notation (e.g., ΔABC ~ ΔDEF). Make sure the corresponding vertices are in the same order.

3. Set Up Proportions: Based on the similarity statement, set up proportions using the corresponding sides of the similar triangles. Remember that corresponding sides are those opposite congruent angles.

4. Solve for Unknowns: Use algebraic methods to solve the proportions for any unknown side lengths.

5. Verify Your Solution: Double-check your work by plugging your solution back into the proportions and making sure they hold true. Consider checking the ratios of the sides to ensure they are consistent with the similarity.

Practice Problems with Detailed Solutions

Let's tackle some practice problems showcasing different aspects of similar triangle applications.

Problem 1: Using AA Similarity

Two triangles, ΔABC and ΔDEF, have the following angles: ∠A = 40°, ∠B = 70°, ∠C = 70°; ∠D = 40°, ∠E = 70°, ∠F = 70°. Are the triangles similar? Explain your reasoning.

Solution:

1. Identify Similar Triangles: We are given the angles of both triangles.

2. State the Similarity: Since ∠A = ∠D = 40° and ∠B = ∠E = 70°, by the AA Similarity Postulate, ΔABC ~ ΔDEF.

3. Proportions (Not needed here): The AA postulate proves similarity without the need for proportions.

4. Solve for Unknowns (Not applicable): No unknowns to solve.

5. Verify: The angles match confirming similarity.

Problem 2: Using SSS Similarity

Two triangles, ΔPQR and ΔXYZ, have sides with the following lengths: PQ = 6, QR = 8, PR = 10; XY = 9, YZ = 12, XZ = 15. Are the triangles similar?

Solution:

1. Identify Similar Triangles: We compare the ratios of corresponding sides.

2. State the Similarity: Let's check the ratios: XY/PQ = 9/6 = 1.5; YZ/QR = 12/8 = 1.5; XZ/PR = 15/10 = 1.5. Since all ratios are equal, the triangles are similar.

3. Proportions: The ratios themselves are the proportions.

4. Solve for Unknowns (Not applicable): No unknowns.

5. Verify: All ratios are equal, confirming similarity. Therefore, ΔPQR ~ ΔXYZ by SSS Similarity Theorem.

Problem 3: Using SAS Similarity and Solving for Unknowns

In ΔABC and ΔDEF, ∠B = ∠E = 55°. AB = 4, BC = 6, DE = 6, EF = 9. Are the triangles similar? If so, find the length of AC.

Solution:

1. Identify Similar Triangles: We have one congruent angle and need to check the ratio of sides around this angle.

2. State the Similarity: Let's check the ratio: AB/DE = 4/6 = 2/3; BC/EF = 6/9 = 2/3. The ratio of corresponding sides around the congruent angles is equal.

3. Proportions: Since AB/DE = BC/EF and ∠B = ∠E, by SAS Similarity, ΔABC ~ ΔDEF.

4. Solve for Unknowns: Now we set up the proportion for AC and DF: AC/DF = 2/3. We need more information to solve for AC. Let's assume we are given DF = 12. Then, AC/12 = 2/3, which means AC = (2/3) * 12 = 8.

5. Verify: Substitute AC = 8 into the proportion: 8/12 = 2/3, which is true.

Problem 4: Similar Triangles in Real-World Applications (Shadow Problem)

A tree casts a shadow of 20 meters. At the same time, a 1.5-meter-tall person casts a shadow of 2 meters. How tall is the tree?

Solution:

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

2. State the Similarity: Let h be the height of the tree. The triangles are similar (AA similarity, as both have a right angle and the angle of the sun is the same for both).

3. Proportions: We set up the proportion: h/20 = 1.5/2

4. Solve for Unknowns: Solving for h: h = (1.5/2) * 20 = 15 meters.

5. Verify: 15/20 = 1.5/2 (Both are 0.75), confirming the solution.

Problem 5: More Complex Problem with Multiple Similar Triangles

In the diagram [imagine a diagram showing two larger similar triangles, each containing a smaller similar triangle within them], ΔABC ~ ΔADE, and ΔADE ~ ΔBGF. If AB = 12, BC = 8, and DE = 6, find the lengths of AD, AE, and BG.

Solution:

This problem requires a sequential approach. Since ΔABC ~ ΔADE, we can use the ratio of corresponding sides: AB/AD = BC/DE = AC/AE.

1. Finding AD and AE: We have AB/AD = BC/DE => 12/AD = 8/6. Solving for AD: AD = (12 * 6) / 8 = 9. Similarly, to find AE, use AB/AD = AC/AE. We need to find AC first (Pythagorean theorem on ΔABC). AC² = AB² + BC² = 12² + 8² = 208. AC = √208 ≈ 14.42. Then, 12/9 = 14.42/AE. Therefore AE ≈ 10.81.

2. Finding BG: Since ΔADE ~ ΔBGF, we can use the ratio of corresponding sides again: AD/BG = DE/GF = AE/BF. We need more information or relationships to find BG. Assume, for instance, that we know GF = 4. Then, 9/BG = 6/4, leading to BG = 6.

Frequently Asked Questions (FAQ)

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

  • A: Congruent triangles are identical in shape and size, while similar triangles have the same shape but may differ in size. Congruent triangles have all corresponding sides and angles equal, whereas similar triangles have proportional sides and equal angles.

• Q: Can any two triangles be similar?

  • A: No, only triangles that meet the conditions of AA, SSS, or SAS similarity can be declared similar.

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

  • A: Corresponding sides are opposite the congruent angles. The order of vertices in the similarity statement (e.g., ΔABC ~ ΔDEF) indicates the correspondence: A corresponds to D, B to E, and C to F.

Conclusion

Mastering similar triangles involves understanding the underlying theorems and developing a systematic approach to problem-solving. By applying the AA, SSS, and SAS similarity theorems and practicing with various problems, you can confidently tackle complex geometric scenarios. Remember to always check your solutions and consider real-world applications to deepen your understanding and appreciation of this fundamental geometric concept. The ability to identify and utilize similar triangles is a valuable skill that extends far beyond the classroom, impacting diverse fields where spatial reasoning and precise calculations are crucial.