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Chapter Analysis
Intermediate24 pages • EnglishQuick Summary
In Chapter 6 of the Class 10 Mathematics NCERT textbook, the concept of similarity of triangles is thoroughly explored. The chapter introduces the criteria for the similarity of triangles, such as AA (Angle-Angle), SSS (Side-Side-Side), and SAS (Side-Angle-Side). Various theorems and proofs are provided to reinforce these concepts, including the Basic Proportionality Theorem (Thales Theorem). Through exercises and examples, students learn to apply these criteria to solve problems involving similar triangles.
Key Topics
- •Similarity of figures
- •AAA criterion of similarity
- •SSS Criterion of Similarity
- •SAS Criterion of Similarity
- •Basic Proportionality Theorem
- •Thales Theorem
- •Applications of Similarity
- •Criteria for similarity of triangles
Learning Objectives
- ✓Understand and apply the concepts of similar figures and polygons.
- ✓Learn and utilize the criteria for the similarity of triangles.
- ✓Prove theorems relating to similarity, including the Basic Proportionality Theorem.
- ✓Apply similarity criteria to solve geometric problems involving triangles.
- ✓Distinguish between congruence and similarity and apply relevant criteria appropriately.
- ✓Develop problem-solving skills using geometric concepts.
Questions in Chapter
State which pairs of triangles in Fig. 6.34 are similar. Write the similarity criterion used by you for answering the question and also write the pairs of similar triangles in the symbolic form.
Page 95
In Fig. 6.35, ΔODC ~ ΔOBA, ∠BOC = 125° and ∠CDO = 70°. Find ∠DOC, ∠DCO, and ∠OAB.
Page 96
Diagonals AC and BD of a trapezium ABCD with AB || DC intersect each other at the point O. Using a similarity criterion for two triangles, show that OA/OB = OC/OD.
Page 96
In Fig. 6.36, QR/QT = QS/PR and ∠1 = ∠2. Show that ΔPQS ~ ΔTQR.
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S and T are points on sides PR and QR of ΔPQR such that ∠P = ∠RTS. Show that ΔRPQ ~ ΔRTS.
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In Fig. 6.37, if ΔABE ≅ ΔACD, show that ΔADE ~ ΔABC.
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In Fig. 6.38, altitudes AD and CE of ΔABC intersect each other at the point P. Show that: (i) ΔAEP ~ ΔCDP (ii) ΔABD ~ ΔCBE (iii) ΔAEP ~ ΔADB (iv) ΔPDC ~ ΔBEC.
Page 97
E is a point on the side AD produced of a parallelogram ABCD and BE intersects CD at F. Show that ΔABE ~ ΔCFB.
Page 97
Additional Practice Questions
Given two triangles ABC and DEF, if ∠A = ∠D and AB/DE = AC/DF, are the triangles similar?
mediumAnswer: Yes, by the SAS similarity criterion, the triangles ABC and DEF are similar.
What condition should two triangles meet for them to be considered similar using the AA criterion?
easyAnswer: Two triangles are considered similar using the AA criterion if two angles of one triangle are equal to two angles of the other triangle.
Explain how the Basic Proportionality Theorem can be applied to solve real-world problems.
hardAnswer: The Basic Proportionality Theorem can be applied in situations where structures are similar, such as scaling models. It helps in determining unknown lengths when parallel lines divide two sides of a triangle proportionally, aiding in architecture and engineering.
If triangles ABC and XYZ are similar and BC = 8 cm, YZ = 4 cm, and ∠A = ∠X, find the ratio of the areas of these triangles.
mediumAnswer: The ratio of the areas of two similar triangles is equal to the square of the ratio of their corresponding sides. Therefore, the ratio of the areas of ΔABC and ΔXYZ is (BC/YZ)^2 = (8/4)^2 = 4.
In a right-angled triangle, if the hypotenuse and one side of one triangle are respectively proportional to the hypotenuse and one side of another triangle, what can be concluded about the triangles?
mediumAnswer: If the hypotenuse and one side of a right triangle are proportional to those of another, the two triangles are similar by the RHS criterion.