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Triangles

Triangles

Action PlanNotesQuestionsMock-Test
Study Action Plan (Sorted by Difficulty)
Proof-based Applications of BPT
Accuracy: 43%
Total Attempted70
Total Correct30
Total Incorrect40
Difficulty Level:
High
RHS Similarity Criterion
Accuracy: 73%
Total Attempted26
Total Correct19
Total Incorrect7
Difficulty Level:
Medium
Basic Proportionality Theorem (Thales Theorem)
Accuracy: 78%
Total Attempted139
Total Correct108
Total Incorrect31
Difficulty Level:
Low
Correct Correspondence in Similar Triangles
Accuracy: 78%
Total Attempted45
Total Correct35
Total Incorrect10
Difficulty Level:
Low
Medians / Altitudes / Angle Bisectors in Similar Triangles
Accuracy: 79%
Total Attempted222
Total Correct176
Total Incorrect46
Difficulty Level:
Low
Applications of Similar Triangles
Accuracy: 79%
Total Attempted155
Total Correct123
Total Incorrect32
Difficulty Level:
Low
Similarity of Triangles
Accuracy: 81%
Total Attempted283
Total Correct228
Total Incorrect55
Difficulty Level:
Low
Converse of Basic Proportionality Theorem
Accuracy: 87%
Total Attempted95
Total Correct83
Total Incorrect12
Difficulty Level:
Low

Learning Objectives

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Learning Objectives

  • Understand the conditions under which two triangles are similar, focusing on the AAA, AA, SSS, SAS, and RHS similarity criteria.
  • Calculate the ratios of corresponding sides in similar triangles using the Basic Proportionality Theorem (Thales Theorem).
  • Apply the converse of the Basic Proportionality Theorem to determine parallel lines in a triangle.
  • Solve problems involving the similarity of triangles using the criteria for similarity, including AAA, AA, SSS, SAS, and RHS.
  • Analyze geometric figures to identify similar triangles and use similarity to solve real-world problems, such as indirect measurements of heights and distances.
  • Prove the similarity of triangles using given geometric conditions and theorems, focusing on the use of parallel lines and proportional segments.

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Revision Notes & Summary

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Chapter Notes

Similarity of Triangles

  • Definition: Two triangles are similar if:
    • Their corresponding angles are equal.
    • Their corresponding sides are in the same ratio.
  • Criteria for Similarity:
    • AAA Criterion: If all three angles of one triangle are equal to all three angles of another triangle, the triangles are similar.
    • AA Criterion: If two angles of one triangle are equal to two angles of another triangle, the triangles are similar.
    • SSS Criterion: If the sides of one triangle are proportional to the sides of another triangle, the triangles are similar.
    • SAS Criterion: If one angle of a triangle is equal to one angle of another triangle and the sides including these angles are proportional, the triangles are similar.
    • RHS Criterion: In right triangles, if the hypotenuse and one side of one triangle are proportional to the hypotenuse and one side of another triangle, the triangles are similar.

Basic Proportionality Theorem (Thales Theorem)

  • Statement: If a line is drawn parallel to one side of a triangle to intersect the other two sides, the other two sides are divided in the same ratio.

Converse of Basic Proportionality Theorem

  • Statement: If a line divides two sides of a triangle in the same ratio, then the line is parallel to the third side.

Examples and Applications

  • Example 1: If a line intersects sides ABAB and ACAC of a triangle △ABC\triangle ABC at DD and EE respectively and is parallel to BCBC, then ADDB=AEEC\frac{AD}{DB} = \frac{AE}{EC}.
  • Example 2: In a trapezium ABCDABCD with AB∥DCAB \parallel DC, if EF∥ABEF \parallel AB, then AEED=BFFC\frac{AE}{ED} = \frac{BF}{FC}.

Exercises

  • Exercise 6.1: Identify similar figures and apply the similarity criteria.
  • Exercise 6.2: Use theorems to prove parallel lines and proportional segments.
  • Exercise 6.3: Solve problems involving medians and altitudes in similar triangles.

Summary

  • Similar Figures: Figures with the same shape but not necessarily the same size.
  • Congruent Figures: All congruent figures are similar, but not all similar figures are congruent.
  • Polygons: Two polygons are similar if their corresponding angles are equal and sides are in proportion.
  • Theorems: Basic Proportionality Theorem and its converse are fundamental in proving similarity in triangles.
  • Similarity Criteria: AAA, AA, SSS, SAS, and RHS provide methods to establish similarity between triangles.
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Practice Test – MCQs, True/False

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Multiple Choice Questions

A.

DEDE is parallel to BCBC.

B.

DEDE is the angle bisector of ∠BAC\angle BAC.

C.

DEDE is the perpendicular bisector of BCBC.

D.

DEDE is a median of △ABC\triangle ABC.
Correct Answer: A

Solution:

By the converse of the Basic Proportionality Theorem, if ADDB=AEEC\frac{AD}{DB} = \frac{AE}{EC}, then DEDE is parallel to BCBC.

Chapter Concept:

Converse of Basic Proportionality Theorem

A.

6 cm

B.

5 cm

C.

4 cm

D.

3 cm
Correct Answer: C

Solution:

For the triangles to be similar by the RHS criterion, the sides must be proportional. The ratio of the hypotenuses is 510=12\frac{5}{10} = \frac{1}{2}. Therefore, the corresponding leg in the second triangle should be 3×2=63 \times 2 = 6 cm.

Chapter Concept:

RHS Similarity Criterion

A.

Pythagorean Theorem

B.

Basic Proportionality Theorem

C.

SAS Similarity Criterion

D.

SSS Similarity Criterion
Correct Answer: B

Solution:

The Basic Proportionality Theorem states that if a line is drawn parallel to one side of a triangle to intersect the other two sides, then it divides those sides in the same ratio.

Chapter Concept:

Proof-based Applications of BPT

A.

2 cm

B.

3 cm

C.

4 cm

D.

6 cm
Correct Answer: B

Solution:

Since EF∥AB∥DCEF \parallel AB \parallel DC, by the Basic Proportionality Theorem, AEED=BFFC\frac{AE}{ED} = \frac{BF}{FC}. Substituting the given values, 32=4x\frac{3}{2} = \frac{4}{x}. Solving for xx, we get x=3 cmx = 3 \text{ cm}.

Chapter Concept:

Proof-based Applications of BPT

A.

10 cm10 \text{ cm}

B.

12 cm12 \text{ cm}

C.

15 cm15 \text{ cm}

D.

16 cm16 \text{ cm}
Correct Answer: B

Solution:

Since △ABC∼△DEF\triangle ABC \sim \triangle DEF, the sides are proportional: ABDE=BCEF=CAFD\frac{AB}{DE} = \frac{BC}{EF} = \frac{CA}{FD}. Solving 5FD=36\frac{5}{FD} = \frac{3}{6} gives FD=10 cmFD = 10 \text{ cm}.

Chapter Concept:

Correct Correspondence in Similar Triangles

A.

8 cm

B.

7.5 cm

C.

6.5 cm

D.

8.5 cm
Correct Answer: A

Solution:

Since △ABC∼△DEF\triangle ABC \sim \triangle DEF, ABDE=BCEF=CADF\frac{AB}{DE} = \frac{BC}{EF} = \frac{CA}{DF}. Substituting the known values, 36=4EF=510\frac{3}{6} = \frac{4}{EF} = \frac{5}{10}. Solving for EFEF, we get EF=8EF = 8 cm.

Chapter Concept:

Similarity of Triangles

A.

The triangles are congruent.

B.

The triangles are similar by SSS criterion.

C.

The triangles are similar by SAS criterion.

D.

The triangles are not similar.
Correct Answer: B

Solution:

The triangles are similar by the SSS (Side-Side-Side) similarity criterion because their corresponding sides are in the same ratio.

Chapter Concept:

Similarity of Triangles

A.

The line is perpendicular to the third side.

B.

The line is parallel to the third side.

C.

The line bisects the third side.

D.

The line is a median of the triangle.
Correct Answer: B

Solution:

According to the converse of the Basic Proportionality Theorem, if a line divides two sides of a triangle in the same ratio, then the line is parallel to the third side.

Chapter Concept:

Converse of Basic Proportionality Theorem

A.

∠A=∠D\angle A = \angle D

B.

∠A=∠E\angle A = \angle E

C.

∠A=∠F\angle A = \angle F

D.

∠A=∠C\angle A = \angle C
Correct Answer: A

Solution:

In similar triangles, corresponding angles are equal. Therefore, ∠A\angle A corresponds to ∠D\angle D, making them equal.

Chapter Concept:

Medians / Altitudes / Angle Bisectors in Similar Triangles

A.

6 cm

B.

9 cm

C.

4 cm

D.

3 cm
Correct Answer: A

Solution:

By the Angle Bisector Theorem, AB/AC=BD/DCAB/AC = BD/DC. So, 6/9=4/x⇒x=6 cm6/9 = 4/x \Rightarrow x = 6 \text{ cm}.

Chapter Concept:

Basic Proportionality Theorem (Thales Theorem)

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True or False

Correct Answer: True

Solution:

Theorem 6.5 states that if one angle of a triangle is equal to one angle of another triangle and the sides including these angles are proportional, then the two triangles are similar. This is the SAS similarity criterion.

Chapter Concept :

Applications of Similar Triangles

Correct Answer: True

Solution:

This statement is true and represents the converse of the Basic Proportionality Theorem.

Chapter Concept :

Basic Proportionality Theorem (Thales Theorem)

Correct Answer: True

Solution:

The Basic Proportionality Theorem, also known as Thales Theorem, confirms that a line parallel to one side of a triangle divides the other two sides proportionally.

Chapter Concept :

Basic Proportionality Theorem (Thales Theorem)

Correct Answer: False

Solution:

The SSS similarity criterion states that if in two triangles, the sides are in the same ratio, the triangles are similar, not necessarily congruent.

Chapter Concept :

Applications of Similar Triangles

Correct Answer: False

Solution:

In the case of similarity of triangles, it is not necessary to check both conditions as one condition implies the other.

Chapter Concept :

Correct Correspondence in Similar Triangles

Correct Answer: False

Solution:

The AAA similarity criterion states that if the corresponding angles of two triangles are equal, then the triangles are similar. The SSS criterion involves corresponding sides being in the same ratio.

Chapter Concept :

Correct Correspondence in Similar Triangles

Correct Answer: True

Solution:

This statement is true and is known as the Converse of the Basic Proportionality Theorem. It states that if a line divides any two sides of a triangle in the same ratio, then the line is parallel to the third side.

Chapter Concept :

Converse of Basic Proportionality Theorem

Correct Answer: True

Solution:

This is the SSS (Side-Side-Side) similarity criterion, which states that if the sides of one triangle are proportional to the sides of another, their corresponding angles are equal, and hence, the triangles are similar.

Chapter Concept :

Similarity of Triangles

Correct Answer: True

Solution:

This statement is true and is known as the Basic Proportionality Theorem or Thales Theorem.

Chapter Concept :

Basic Proportionality Theorem (Thales Theorem)

Correct Answer: True

Solution:

This statement is the converse of the Basic Proportionality Theorem, which states that if a line divides any two sides of a triangle in the same ratio, then the line is parallel to the third side.

Chapter Concept :

Converse of Basic Proportionality Theorem

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