Exercise 6.3 Class 10 Solutions

Exercise 6.3 Class 10 Solutions: A complete walkthrough to Understanding Triangles

This article provides comprehensive solutions and explanations for Exercise 6.3 in Class 10 mathematics textbooks, typically covering theorems related to similar triangles. Understanding this exercise is crucial for mastering concepts of similarity, proportionality, and their applications in geometry. We'll break down each problem step-by-step, clarifying the underlying principles and offering additional insights to solidify your understanding. This guide aims to not only provide answers but also encourage a deeper comprehension of similar triangles.

Introduction to Similar Triangles and Key Theorems

Before diving into the solutions, let's refresh our understanding of similar triangles. Two triangles are considered similar if their corresponding angles are equal and their corresponding sides are proportional. This proportionality is often expressed as a ratio.

• AAA (Angle-Angle-Angle) Similarity: If all three angles of one triangle are equal to the three corresponding angles of another triangle, the triangles are similar.

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

• SSS (Side-Side-Side) Similarity: If the corresponding sides of two triangles are proportional, the triangles are similar.

• SAS (Side-Angle-Side) Similarity: If two pairs of corresponding sides of two triangles are proportional, and the included angle between those sides is equal, the triangles are similar.

These theorems are the cornerstones of solving problems in Exercise 6.3 and similar exercises dealing with similar triangles. Remember that understanding the why behind each step is as important as getting the correct answer.

Detailed Solutions to Exercise 6.3 Class 10 Problems

(Note: The specific problems in Exercise 6.3 will vary depending on the textbook used. This section provides example problems and their solutions to illustrate the application of the above theorems. Replace these examples with the actual problems from your textbook.)

Example Problem 1:

Problem: In ΔABC and ΔDEF, ∠A = ∠D = 60°, ∠B = ∠E = 80°, and ∠C = ∠F = 40°. Prove that ΔABC ~ ΔDEF.

Solution:

This problem directly applies the AAA similarity theorem. We are given that:

• ∠A = ∠D = 60°
• ∠B = ∠E = 80°
• ∠C = ∠F = 40°

Since all three corresponding angles are equal, by the AAA similarity criterion, we conclude that ΔABC ~ ΔDEF.

Example Problem 2:

Problem: In ΔPQR and ΔXYZ, PQ/XY = QR/YZ = RP/ZX = 2/3. Prove that ΔPQR ~ ΔXYZ.

Solution:

This problem uses the SSS similarity theorem. We are given that the ratio of corresponding sides is constant:

• PQ/XY = 2/3
• QR/YZ = 2/3
• RP/ZX = 2/3

Since all three pairs of corresponding sides are proportional, by the SSS similarity criterion, we conclude that ΔPQR ~ ΔXYZ. Simple as that.

Example Problem 3:

Problem: In ΔABC, DE || BC. If AD = 2 cm, DB = 3 cm, and AE = 4 cm, find EC.

Solution:

This problem utilizes the concept of Basic Proportionality Theorem (Thales' Theorem), which states that if a line is drawn parallel to one side of a triangle intersecting the other two sides, then it divides the other two sides proportionally. Since DE || BC, we have:

AD/DB = AE/EC

Substituting the given values:

2/3 = 4/EC

Cross-multiplying:

2 * EC = 3 * 4

2EC = 12

EC = 6 cm

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That's why, EC = 6 cm.

Example Problem 4 (Involving Area):

Problem: Two triangles are similar. The ratio of their corresponding sides is 3:5. Find the ratio of their areas.

Solution:

The ratio of the areas of two similar triangles is equal to the square of the ratio of their corresponding sides. Since the ratio of their sides is 3:5, the ratio of their areas is:

(3/5)² = 9/25

So, the ratio of their areas is 9:25.

Example Problem 5 (More Complex):

Problem: In a triangle ABC, D and E are points on AB and AC respectively such that DE || BC. If AD = x, DB = x+2, AE = x-2, and EC = x-1, find the value of x.

Solution:

Using the Basic Proportionality Theorem:

AD/DB = AE/EC

Substituting the given values:

x/(x+2) = (x-2)/(x-1)

Cross-multiplying:

x(x-1) = (x+2)(x-2)

x² - x = x² - 4

-x = -4

x = 4

That's why, the value of x is 4.

Explanation of Underlying Principles

The solutions above demonstrate the application of several key geometric principles:

• Similarity: The core concept is understanding what constitutes similar triangles. The equal angles and proportional sides are key.

• Proportionality: Many problems involve setting up and solving proportions. This requires a solid understanding of ratios and cross-multiplication.

• Basic Proportionality Theorem (Thales' Theorem): This theorem is crucial for problems involving parallel lines intersecting the sides of a triangle.

• Area Relationships in Similar Triangles: The relationship between the ratio of sides and the ratio of areas is essential for certain problems.

Frequently Asked Questions (FAQ)

• Q: What if I don't remember all the similarity theorems?

A: Focus on understanding the underlying concepts of equal angles and proportional sides. Even if you don't recall the specific names, you can still solve many problems by applying these fundamental principles. Reviewing the theorems regularly will improve your recall.

• Q: How can I improve my problem-solving skills in this area?

A: Practice is key! Don't just look for the answers; try to solve each problem yourself first. Work through numerous examples. If you get stuck, carefully review the steps and understand where you went wrong.

• Q: Are there any shortcut methods for solving these problems?

A: While there aren't shortcuts that bypass understanding the theorems, recognizing patterns and common problem types can help you solve problems more efficiently. The more problems you solve, the better you'll become at identifying the best approach.

• Q: What resources can I use to learn more about similar triangles?

A: Your textbook, class notes, and online resources (like educational websites and videos) are all excellent places to find more information and practice problems.

Conclusion

Mastering Exercise 6.On top of that, 3 and the broader topic of similar triangles requires a solid grasp of the underlying theorems and a willingness to practice consistently. Remember, the key is not just finding the answer but understanding the why behind each step, ensuring a deep and lasting comprehension of these important geometric concepts. By understanding the principles of similarity, proportionality, and the Basic Proportionality Theorem, you can confidently tackle even the most challenging problems. This understanding will serve as a strong foundation for more advanced mathematical studies.