If two triangles $\triangle ABC$ and $\triangle DEF$ are similar with $\frac{AB}{DE} = \frac{BC}{EF} = \frac{CA}{FD} = 2$, and $DE = 3$ cm, what is the length of $AB$?

Correct Option: B. Solution: Since $\frac{AB}{DE} = 2$ and $DE = 3$ cm, we have $AB = 2 \times 3 = 6$ cm.

Consider two triangles $\triangle PQR$ and $\triangle XYZ$ such that $\frac{PQ}{XY} = \frac{QR}{YZ} = \frac{RP}{ZX} = 2$. If $PQ = 4$ cm, what is the length of $XY$?

Correct Option: A. Solution: Since $\frac{PQ}{XY} = 2$, we have $XY = \frac{PQ}{2} = \frac{4}{2} = 2$ cm.

Triangles

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Summary

Summary of Triangles

Introduction to Triangles

Congruent figures have the same shape and size.
Similar figures have the same shape but not necessarily the same size.

Similarity Criteria for Triangles

SSS Similarity Criterion: If corresponding sides of two triangles are in the same ratio, then the triangles are similar.
SAS Similarity Criterion: If one angle of a triangle is equal to one angle of another triangle and the sides including these angles are in the same ratio, then the triangles are similar.
RHS Similarity Criterion: In right triangles, if the hypotenuse and one side of one triangle are proportional to the hypotenuse and one side of another triangle, then the triangles are similar.

Properties of Similar Figures

All circles with the same radius are congruent.
All squares with the same side lengths are congruent.
All equilateral triangles with the same side lengths are congruent.
Similar figures have equal corresponding angles and proportional corresponding sides.

Applications of Similarity

Used in indirect measurements, such as calculating heights of mountains or distances to celestial objects.

Important Theorems

Theorem 6.1: If a line divides two sides of a triangle in the same ratio, it is parallel to the third side.
Theorem 6.2: A line drawn through the mid-point of one side of a triangle parallel to another side bisects the third side.

Example Applications

Proving similarity in various geometric configurations using the criteria mentioned above.

Learning Objectives

Understand the concept of similarity in triangles.
Identify and apply the SSS similarity criterion for triangles.
Identify and apply the SAS similarity criterion for triangles.
Understand the RHS similarity criterion for right triangles.
Prove the similarity of triangles using given conditions and properties.
Apply theorems related to parallel lines and proportional segments in triangles.
Solve problems involving indirect measurements using the principles of similarity.

Detailed Notes

Chapter Notes on Triangles

6.1 Introduction

Congruence of Triangles: Two figures are congruent if they have the same shape and size.
Similarity of Figures: Figures that have the same shape but not necessarily the same size are called similar figures.
Application: Similarity of triangles is used in indirect measurements, such as finding heights of mountains or distances to the moon.

6.2 Similar Figures

Definition: All circles with the same radii are congruent, all squares with the same side lengths are congruent, and all equilateral triangles with the same side lengths are congruent.
Similar Figures: All circles, squares, and equilateral triangles are similar as they have the same shape.
Congruent vs Similar: All congruent figures are similar, but similar figures need not be congruent.

6.3 Criteria for Similarity of Triangles

SSS Similarity Criterion: If in two triangles, corresponding sides are in the same ratio, then their corresponding angles are equal, and hence the triangles are similar.
SAS Similarity Criterion: If one angle of a triangle is equal to one angle of another triangle and the sides including these angles are in the same ratio, then the triangles are similar.
RHS Similarity Criterion: In two right triangles, if the hypotenuse and one side of one triangle are proportional to the hypotenuse and one side of the other triangle, then the triangles are similar.

6.4 Examples and Theorems

Example: If a line intersects sides AB and AC of a triangle at points D and E respectively and is parallel to BC, then the ratios of the segments are equal:
$\frac{AD}{DB} = \frac{AE}{EC}$
Theorem 6.1: If a line divides any two sides of a triangle in the same ratio, then the line is parallel to the third side.
Theorem 6.2: A line drawn through the midpoint of one side of a triangle parallel to another side bisects the third side.

6.5 Applications of Similarity

Indirect Measurement: Similarity is used to find heights and distances indirectly.
Example Problem: A girl of height 90 cm walking away from a lamp-post of height 3.6 m can have the length of her shadow calculated using the principles of similarity.

Important Diagrams

Fig. 6.36: Shows triangles with corresponding sides and angles.
Fig. 6.37: Illustrates altitudes intersecting in triangles.
Fig. 6.38: Depicts right-angled triangles and their properties.
Fig. 6.39: Demonstrates angle bisectors and their relationships in similar triangles.

Exam Tips & Common Mistakes

Common Mistakes and Exam Tips

Common Pitfalls

Misidentifying Similar Triangles: Students often confuse congruent triangles with similar triangles. Remember, similar triangles have the same shape but not necessarily the same size.
Incorrect Application of Similarity Criteria: Ensure you apply the correct criteria for similarity (SSS, SAS, AA, etc.) when proving triangles are similar.
Ignoring Angle Relationships: Failing to recognize that if two angles are equal, the triangles are similar can lead to incorrect conclusions.

Tips for Success

Always Check Ratios: When determining if triangles are similar, check that the ratios of corresponding sides are equal.
Use Diagrams: Draw clear diagrams to visualize the relationships between triangles and their corresponding parts.
Practice with Examples: Work through multiple examples to familiarize yourself with different scenarios where similarity applies.
Label Corresponding Parts: Clearly label corresponding angles and sides when proving similarity to avoid confusion.
Review Theorems: Make sure you understand and can apply theorems related to triangle similarity, such as the theorem stating that a line dividing two sides of a triangle in the same ratio is parallel to the third side.

Practice & Assessment

Multiple Choice Questions

All squares are congruent.

All squares are similar.

All squares are equilateral.

All squares have different angles.

Correct Answer: B

Solution:

All squares are similar because they have the same shape but not necessarily the same size.

Rectangle

Rhombus

Parallelogram

Kite

Correct Answer: D

Solution:

The quadrilateral has two pairs of equal adjacent sides, which is characteristic of a kite.

\triangle DEF

is congruent to

\triangle ABC

\triangle DEF

is similar to

\triangle ABC

\triangle DEF

is half the area of

\triangle ABC

\triangle DEF

is one-fourth the area of

\triangle ABC

Correct Answer: D

Solution:

\triangle DEF

is similar to

\triangle ABC

and is one-fourth the area of

\triangle ABC

because it is formed by joining the midpoints of the sides of

\triangle ABC

Yes, by SSS similarity

Yes, by SAS similarity

No, they are not similar

Yes, by AAA similarity

Correct Answer: C

Solution:

For triangles to be similar by SSS, the ratios of corresponding sides must be equal. The ratios

\frac{LM}{DE} = \frac{3}{4}

\frac{MP}{EF} = \frac{2}{5}

, and

\frac{PL}{FD} = \frac{2.7}{6}

are not equal, so the triangles are not similar.

60°

80°

100°

120°

Correct Answer: B

Solution:

Using the angle sum property of triangles,

\angle A = 180^\circ - 60^\circ - 40^\circ = 80^\circ

1.2 m

2.4 m

3.6 m

4.8 m

Correct Answer: A

Solution:

After 4 seconds, the girl is 4.8 m away from the lamp-post. Using similar triangles,

\frac{90}{360} = \frac{x}{4.8}

, solving gives x = 1.2 m.

Triangles are congruent.

Triangles have equal perimeters.

Corresponding angles are equal.

Triangles have equal areas.

Correct Answer: C

Solution:

Similar triangles have corresponding angles equal and sides in proportion.

1:1

2:1

4:1

3:1

Correct Answer: C

Solution:

The area of a triangle is proportional to the square of its sides when the included angle is the same. Thus, the ratio of the areas is the square of the ratio of the sides, which is 4:1.

3:4

9:16

12:16

16:9

Correct Answer: B

Solution:

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 is

3^2:4^2 = 9:16

\frac{AO}{OC} = \frac{BO}{OD}

\frac{AO}{OD} = \frac{BO}{OC}

\frac{AO}{BO} = \frac{OC}{OD}

\frac{AO}{OC} = \frac{OD}{BO}

Correct Answer: A

Solution:

In a trapezium with diagonals intersecting, the segments of the diagonals are proportional:

\frac{AO}{OC} = \frac{BO}{OD}

. Substituting the given values:

\frac{3}{4} = \frac{2}{5}

, confirming the proportionality.

Two triangles are similar if all corresponding angles are equal.

Two triangles are similar if all corresponding sides are equal.

Two triangles are similar if they have the same area.

Two triangles are similar if they have one pair of equal angles.

Correct Answer: A

Solution:

Two triangles are similar if all corresponding angles are equal and the sides are in proportion.

9:25

3:5

5:3

15:9

Correct Answer: B

Solution:

The ratio of the perimeters of similar triangles is the same as the ratio of their corresponding sides, which is 3:5.

3 cm

6 cm

9 cm

12 cm

Correct Answer: B

Solution:

Since

\frac{AB}{DE} = 2

and

DE = 3

cm, we have

AB = 2 \times 3 = 6

cm.

It bisects the third side.

It is perpendicular to the third side.

It forms an angle of 45° with the third side.

It does not affect the third side.

Correct Answer: A

Solution:

According to the Midpoint Theorem, a line drawn through the midpoint of one side of a triangle parallel to another side bisects the third side.

They are congruent.

They are similar.

They are isosceles.

They are equilateral.

Correct Answer: B

Solution:

If two triangles have their corresponding sides in the same ratio, they are similar by the SSS similarity criterion.

Their corresponding angles are equal.

Their corresponding sides are equal.

Their areas are equal.

Their perimeters are equal.

Correct Answer: A

Solution:

For two triangles to be similar, their corresponding angles must be equal.

8 cm

12 cm

6 cm

10 cm

Correct Answer: B

Solution:

Since

DE \parallel BC

, by the Basic Proportionality Theorem (also known as Thales' theorem),

\frac{AD}{DB} = \frac{DE}{BC}

. Substituting the given values:

\frac{3}{6} = \frac{4}{BC}

. Solving for

BC

gives

BC = 8

cm.

\triangle ABC \sim \triangle PQR

\triangle ABC \cong \triangle PQR

\triangle ABC \not\sim \triangle PQR

None of the above

Correct Answer: A

Solution:

Triangles are similar if their corresponding angles are equal. Since all corresponding angles are equal,

\triangle ABC \sim \triangle PQR

by the AA similarity criterion.

3 cm

2 cm

4 cm

5 cm

Correct Answer: A

Solution:

Since

DE \parallel BC

, by the Basic Proportionality Theorem (also known as Thales' theorem),

\frac{AD}{DB} = \frac{DE}{BC}

. Substituting the given values,

\frac{3}{6} = \frac{DE}{9}

. Solving for

DE

, we get

DE = 3

cm.

Pythagorean Theorem

Basic Proportionality Theorem (Thales' Theorem)

Angle Bisector Theorem

Midpoint Theorem

Correct Answer: B

Solution:

The Basic Proportionality Theorem states that if a line is drawn parallel to one side of a triangle, it divides the other two sides proportionally.

A triangle with angles

60^\circ

80^\circ

, and

40^\circ

A triangle with angles

50^\circ

70^\circ

, and

60^\circ

A triangle with angles

60^\circ

70^\circ

, and

50^\circ

A triangle with angles

90^\circ

45^\circ

, and

45^\circ

Correct Answer: A

Solution:

Triangles are similar if their corresponding angles are equal. Thus, a triangle with angles

60^\circ

80^\circ

, and

40^\circ

is similar to

\triangle ABC

10 cm

12 cm

14 cm

16 cm

Correct Answer: A

Solution:

Using the Pythagorean theorem,

AC = \sqrt{AB^2 + BC^2} = \sqrt{6^2 + 8^2} = \sqrt{36 + 64} = \sqrt{100} = 10

cm.

2.5 m

3 m

3.5 m

4 m

Correct Answer: B

Solution:

After 5 seconds, the girl is 5 m away from the lamp post. Using similar triangles,

\frac{1.5}{4.5} = \frac{x}{5+x}

, where

x

is the shadow length. Solving gives

x = 3

2 cm

3 cm

1.5 cm

4 cm

Correct Answer: A

Solution:

Since DE is parallel to BC, triangles ADE and ABC are similar. Therefore, the ratio of corresponding sides is equal:

\frac{AD}{DB} = \frac{DE}{EC}

. Substituting the values,

\frac{1.5}{3} = \frac{1}{EC}

, solving gives EC = 2 cm.

2.7 cm

5.4 cm

7.2 cm

9 cm

Correct Answer: D

Solution:

Since

DE \parallel BC

, by the Basic Proportionality Theorem (Thales' theorem), we have

\frac{AD}{DC} = \frac{DE}{BC}

. Substituting the given values,

\frac{7.2}{5.4} = \frac{1.8}{x}

, solving for

x

gives

x = 9

cm.

4 cm

5 cm

6 cm

3 cm

Correct Answer: A

Solution:

Since

\angle P = \angle RTS

, by the Angle Bisector Theorem,

\frac{PS}{SR} = \frac{QT}{TR}

. Substituting the given values:

\frac{3}{6} = \frac{4}{8}

. Therefore,

ST = 4

cm.

AO = CO

BO = DO

\frac{AO}{CO} = \frac{BO}{DO}

\angle AOB = \angle COD

Correct Answer: C

Solution:

Since

\triangle AOB \sim \triangle COD

, the corresponding sides are proportional, so

\frac{AO}{CO} = \frac{BO}{DO}

Yes, because

\frac{PE}{EQ} = \frac{PF}{FR}

No, because

\frac{PE}{EQ} \neq \frac{PF}{FR}

Yes, because

\frac{PE}{PF} = \frac{EQ}{FR}

No, because

\frac{PE}{PF} \neq \frac{EQ}{FR}

Correct Answer: A

Solution:

For

EF

to be parallel to

QR

, the segments must be proportional:

\frac{PE}{EQ} = \frac{PF}{FR}

. Here,

\frac{3.9}{3} = \frac{3.6}{2.4}

, both equal

1.3

, confirming parallelism.

1.5

0.5

Correct Answer: A

Solution:

The scale factor is the ratio of the corresponding sides. If a side of triangle ABC is 3 and the corresponding side of triangle DEF is 6, the scale factor is

\frac{6}{3} = 2

48 cm

72 cm

96 cm

120 cm

Correct Answer: B

Solution:

The perimeter of

\triangle DEF

is twice that of

\triangle ABC

. The perimeter of

\triangle ABC

is 24 cm, so the perimeter of

\triangle DEF

is 48 cm.

The triangles are congruent.

The triangles are similar.

The triangles are neither similar nor congruent.

The triangles are isosceles.

Correct Answer: B

Solution:

According to the SAS similarity criterion, if one angle of a triangle is equal to one angle of another triangle and the sides including these angles are in the same ratio, then the triangles are similar.

4 cm

2 cm

6 cm

8 cm

Correct Answer: A

Solution:

Since DE is parallel to BC, triangles ADE and ABC are similar. Thus, AD/DB = DE/EC, which gives EC = 4 cm.

72 cm

54 cm

90 cm

60 cm

Correct Answer: C

Solution:

The perimeter of a similar triangle scales by the same factor as the sides. The perimeter of

\triangle ABC

6 + 8 + 10 = 24

cm. Scaling by a factor of 3 gives

24 \times 3 = 72

cm.

Square

Rectangle

Rhombus

Irregular quadrilateral

Correct Answer: D

Solution:

The side lengths are not equal, and the angles are not right angles, indicating that

PQRS

is an irregular quadrilateral.

12 cm

14 cm

15 cm

16 cm

Correct Answer: B

Solution:

Since

\triangle PQR \sim \triangle XYZ

, the sides are proportional:

\frac{PQ}{XY} = \frac{QR}{YZ}

. Given

PQ = 5

cm,

XY = 10

cm, and

QR = 6

cm, we have

\frac{5}{10} = \frac{6}{YZ}

. Solving for

YZ

, we get

YZ = 12

cm.

Triangles ADE and ABC are similar.

Triangles ADE and ABC are congruent.

Triangles ADE and ABC are neither similar nor congruent.

Triangles ADE and ABC are identical.

Correct Answer: A

Solution:

Since DE is parallel to BC and the sides are in proportion, triangles ADE and ABC are similar by the Basic Proportionality Theorem.

Yes, by SSS similarity criterion.

No, because the sides are not proportional.

Yes, by SAS similarity criterion.

No, because the angles are not equal.

Correct Answer: A

Solution:

The sides of the triangles are proportional:

\frac{AB}{PQ} = \frac{BC}{QR} = \frac{AC}{PR}

. Therefore, the triangles are similar by the SSS similarity criterion.

All circles are congruent.

All circles are similar.

All circles have the same radius.

All circles are equilateral.

Correct Answer: B

Solution:

All circles are similar because they have the same shape but not necessarily the same size.

\triangle POQ \sim \triangle SOR

by AAA similarity criterion.

\triangle POQ \cong \triangle SOR

by SSS congruence criterion.

\triangle POQ \sim \triangle SOR

by SAS similarity criterion.

\triangle POQ \cong \triangle SOR

by ASA congruence criterion.

Correct Answer: A

Solution:

Since

PQ \parallel RS

\angle P = \angle S

and

\angle Q = \angle R

(alternate angles), and

\angle POQ = \angle SOR

(vertically opposite angles),

\triangle POQ \sim \triangle SOR

by AAA similarity criterion.

The triangles are congruent.

The triangles are similar.

The triangles are neither similar nor congruent.

The triangles are isosceles.

Correct Answer: B

Solution:

The triangles are similar by the Angle-Angle (AA) similarity criterion as the corresponding angles are equal and the sides are proportional.

2 cm

4 cm

8 cm

6 cm

Correct Answer: A

Solution:

Since

\frac{PQ}{XY} = 2

, we have

XY = \frac{PQ}{2} = \frac{4}{2} = 2

cm.

3:4

9:16

4:3

16:9

Correct Answer: A

Solution:

The ratio of the areas of similar triangles is the square of the ratio of their corresponding sides. Therefore, the ratio of the sides is the square root of 9:16, which is 3:4.

\frac{OA}{OC} = \frac{OB}{OD}

\frac{OA}{OB} = \frac{OC}{OD}

\frac{OA}{OD} = \frac{OB}{OC}

\frac{OA}{OC} = \frac{OD}{OB}

Correct Answer: A

Solution:

In a trapezium with parallel sides, the diagonals divide each other proportionally, hence

\frac{OA}{OC} = \frac{OB}{OD}

12 cm extsuperscript{2}

6 cm extsuperscript{2}

24 cm extsuperscript{2}

48 cm extsuperscript{2}

Correct Answer: A

Solution:

The area ratio of similar triangles is the square of the scale factor. Thus, the area of

\triangle PQR

\frac{1}{4}

\triangle ABC

, which is 12 cm extsuperscript{2}.

All equilateral triangles are similar.

All equilateral triangles are congruent.

All equilateral triangles are neither similar nor congruent.

All equilateral triangles are identical.

Correct Answer: A

Solution:

All equilateral triangles are similar because they have equal corresponding angles and their sides are in proportion.

5 cm

6 cm

7 cm

8 cm

Correct Answer: A

Solution:

In a right triangle with sides 3 cm and 4 cm, the hypotenuse is 5 cm, as it forms a 3-4-5 Pythagorean triplet.

3 cm

4 cm

6 cm

9 cm

Correct Answer: D

Solution:

Since

PQ \parallel EF

, by the Basic Proportionality Theorem,

\frac{DP}{PE} = \frac{PQ}{EF}

. Substituting the given values,

\frac{3}{6} = \frac{2}{EF}

. Solving for

EF

, we get

EF = 9

cm.

AA (Angle-Angle)

SSS (Side-Side-Side)

SAS (Side-Angle-Side)

RHS (Right angle-Hypotenuse-Side)

Correct Answer: B

Solution:

The SSS (Side-Side-Side) similarity criterion states that if the corresponding sides of two triangles are in proportion, then the triangles are similar.

\frac{AO}{OC} = \frac{OD}{OB}

\frac{AO}{OB} = \frac{OD}{OC}

\frac{AO}{OD} = \frac{OB}{OC}

\frac{AO}{OC} = \frac{OB}{OD}

Correct Answer: A

Solution:

In similar triangles, corresponding sides are proportional. Therefore,

\frac{AO}{OC} = \frac{OD}{OB}

3 cm

4 cm

5 cm

6 cm

Correct Answer: D

Solution:

Since

GH \parallel EF

, triangles

\triangle DGH

and

\triangle DEF

are similar. Therefore,

\frac{DG}{DE} = \frac{GH}{EF}

. Given

DG = 4

cm,

DE = 8

cm, and

EF = 6

cm, we have

\frac{4}{8} = \frac{GH}{6}

. Solving for

GH

, we get

GH = 6

cm.

They are congruent.

They are similar.

They are neither similar nor congruent.

They are equilateral.

Correct Answer: B

Solution:

Two triangles with corresponding angles equal are similar by the Angle-Angle (AA) similarity criterion.

Yes, because the segments are proportional.

No, because the segments are not proportional.

Yes, because EF is perpendicular to QR.

No, because EF is not a line segment.

Correct Answer: A

Solution:

For EF to be parallel to QR,

\frac{PE}{EQ} = \frac{PF}{FR}

. Substituting the values,

\frac{3.9}{3} = 1.3

and

\frac{3.6}{2.4} = 1.5

. Since the ratios are not equal, EF is not parallel to QR.

4 cm

5 cm

6 cm

8 cm

Correct Answer: B

Solution:

Since

\triangle ABC \sim \triangle DEF

, the corresponding sides are proportional. Thus,

\frac{AB}{DE} = \frac{BC}{EF}

. Substituting the given values,

\frac{6}{3} = \frac{8}{EF}

. Solving for

EF

, we get

EF = 4

cm.

55°

45°

35°

75°

Correct Answer: A

Solution:

Since

\angle BOC + \angle CDO + \angle DOC = 180^\circ

\angle DOC = 180^\circ - 125^\circ - 70^\circ = 55^\circ

1.2 m

2.4 m

3.6 m

4.8 m

Correct Answer: B

Solution:

The girl walks 4.8 m (1.2 m/s * 4 s) away from the lamp. Using similar triangles, the length of the shadow is

\frac{90}{360} \times 480 = 1.2

EF is parallel to PR.

EF is parallel to QR.

EF is parallel to PQ.

EF is not parallel to any side.

Correct Answer: B

Solution:

Since EF is drawn parallel to QR and the segments are in proportion, EF is parallel to QR by the Converse of the Basic Proportionality Theorem.

1.5 cm

2 cm

3 cm

4 cm

Correct Answer: A

Solution:

Since

DE \parallel BC

, the triangles are similar, and the ratio of corresponding sides is equal. Therefore,

EC = 1.5

cm.

AAA (Angle-Angle-Angle)

SSS (Side-Side-Side)

SAS (Side-Angle-Side)

RHS (Right angle-Hypotenuse-Side)

Correct Answer: A

Solution:

Since all corresponding angles are equal,

\triangle ABC \sim \triangle DEF

by the AAA similarity criterion.

5 cm

6 cm

7 cm

8 cm

Correct Answer: A

Solution:

\triangle ABC

, the sum of angles is

180^\circ

. Given

\angle A = 60^\circ

\angle B = 80^\circ

, and

\angle C = 40^\circ

, the triangle is valid. Since no specific side length relationship is given,

BC

is assumed to be the same as

AB

, hence

BC = 5

cm.

60°, 60°, 60°

45°, 45°, 90°

30°, 60°, 90°

None of the above

Correct Answer: D

Solution:

A triangle with sides in the ratio 2:3:4 cannot be an equilateral, isosceles, or right triangle, thus none of the given angle sets apply.

Yes, because all sides are proportional.

No, because the angles are not given.

Yes, because the shape is a rectangle.

No, because the sides are not equal.

Correct Answer: B

Solution:

Similarity requires both proportional sides and equal corresponding angles, which are not provided.

\triangle ABC \sim \triangle DEF

\triangle ABC \cong \triangle DEF

\triangle ABC \approx \triangle DEF

None of the above

Correct Answer: A

Solution:

By the SSS similarity criterion, if the corresponding sides of two triangles are in the same ratio, then the triangles are similar. Therefore,

\triangle ABC \sim \triangle DEF

AAA

SSS

SAS

RHS

Correct Answer: B

Solution:

The sides of the triangles are in proportion: AB/PQ = BC/QR = AC/PR, so the triangles are similar by the SSS criterion.

AAA (Angle-Angle-Angle)

SSS (Side-Side-Side)

SAS (Side-Angle-Side)

RHS (Right angle-Hypotenuse-Side)

Correct Answer: B

Solution:

The sides of

\triangle ABC

and

\triangle DEF

are in the ratio

\frac{6}{9} = \frac{8}{12} = \frac{10}{15} = \frac{2}{3}

. Therefore, by the SSS similarity criterion,

\triangle ABC \sim \triangle DEF

This is a property of triangles where a line drawn through the mid-point of one side parallel to another side will bisect the third side.

Triangles

AI Learning Assistant

Summary

Summary of Triangles

Introduction to Triangles

Similarity Criteria for Triangles

Properties of Similar Figures

Applications of Similarity

Important Theorems

Example Applications

Learning Objectives

Learning Objectives

Detailed Notes

Chapter Notes on Triangles

6.1 Introduction

6.2 Similar Figures

6.3 Criteria for Similarity of Triangles

6.4 Examples and Theorems

Example: If a line intersects sides AB and AC of a triangle at points D and E respectively and is parallel to BC, then the ratios of the segments are equal:

6.5 Applications of Similarity

Important Diagrams

Exam Tips & Common Mistakes

Common Mistakes and Exam Tips

Common Pitfalls

Tips for Success

Practice & Assessment

Multiple Choice Questions

Q1.Which of the following statements is true for all squares?

Correct Answer: B

Q2.In a quadrilateral ABCD, if AB = 4 cm, BC = 5 cm, CD = 4 cm, and DA = 5 cm, what type of quadrilateral is it?

Correct Answer: D

Q3.In a triangle △ABC\triangle ABC△ABC, the midpoints of sides ABABAB, BCBCBC, and CACACA are DDD, EEE, and FFF respectively. If △DEF\triangle DEF△DEF is formed by joining these midpoints, what can be said about △DEF\triangle DEF△DEF?

Correct Answer: D

Q4.In triangle LMP, LM = 3, MP = 2, PL = 2.7, and in triangle DEF, DE = 4, EF = 5, FD = 6. Are the triangles similar?

Correct Answer: C

Q5.In triangle ABC, if ∠B=60∘\angle B = 60^\circ∠B=60∘, ∠C=40∘\angle C = 40^\circ∠C=40∘, and BC = 3 cm, what is ∠A\angle A∠A?

Correct Answer: B

Q6.If a girl of height 90 cm walks away from a lamp-post 3.6 m high at a speed of 1.2 m/s, what is the length of her shadow after 4 seconds?

Correct Answer: A

Q7.If triangles ABC and DEF are similar and ABDE=BCEF=CAFD\frac{AB}{DE} = \frac{BC}{EF} = \frac{CA}{FD}DEAB​=EFBC​=FDCA​, which of the following statements is true?

Correct Answer: C

Q8.In a triangle, if two sides are in the ratio 2:1 and the included angle is 60°, what is the ratio of the areas of the two triangles formed?

Correct Answer: C

Q9.If △PQR\triangle PQR△PQR is similar to △XYZ\triangle XYZ△XYZ and the ratio of their corresponding sides is 3:43:43:4, what is the ratio of their areas?

Correct Answer: B

Q10.In a trapezium ABCDABCDABCD with AB∥CDAB \parallel CDAB∥CD, diagonals ACACAC and BDBDBD intersect at point OOO. If AO=3AO = 3AO=3 cm, OC=4OC = 4OC=4 cm, BO=2BO = 2BO=2 cm, and OD=5OD = 5OD=5 cm, which of the following is true?

Correct Answer: A

Q11.Which of the following statements is true about the similarity of triangles?

Correct Answer: A

Q12.If two triangles are similar and the ratio of their corresponding sides is 3:5, what is the ratio of their perimeters?

Correct Answer: B

Q13.If two triangles △ABC\triangle ABC△ABC and △DEF\triangle DEF△DEF are similar with ABDE=BCEF=CAFD=2\frac{AB}{DE} = \frac{BC}{EF} = \frac{CA}{FD} = 2DEAB​=EFBC​=FDCA​=2, and DE=3DE = 3DE=3 cm, what is the length of ABABAB?

Correct Answer: B

Q14.If a line is drawn through the midpoint of one side of a triangle parallel to another side, what does it do to the third side?

Correct Answer: A

Q15.If two triangles have their corresponding sides in the same ratio, what can be said about the triangles?

Correct Answer: B

Q16.Which of the following statements is true for two similar triangles?

Correct Answer: A

Q17.In a triangle △ABC\triangle ABC△ABC, a line DEDEDE is drawn parallel to the base BCBCBC, dividing the triangle into two smaller triangles △ADE\triangle ADE△ADE and △ABC\triangle ABC△ABC. If AD=3AD = 3AD=3 cm, DB=6DB = 6DB=6 cm, and DE=4DE = 4DE=4 cm, what is the length of BCBCBC?

Correct Answer: B

Correct Answer: A

Correct Answer: A

Q20.In a triangle △PQR\triangle PQR△PQR, if a line segment EFEFEF is drawn parallel to the base QRQRQR, and it divides the other two sides proportionally, which theorem is being applied?

Correct Answer: B

Q21.In a triangle △ABC\triangle ABC△ABC, if ∠A=60∘\angle A = 60^\circ∠A=60∘, ∠B=80∘\angle B = 80^\circ∠B=80∘, and ∠C=40∘\angle C = 40^\circ∠C=40∘, which of the following triangles is similar to △ABC\triangle ABC△ABC based on angle criteria?

Correct Answer: A

Q22.In a right triangle △ABC\triangle ABC△ABC, right-angled at BBB, if AB=6AB = 6AB=6 cm and BC=8BC = 8BC=8 cm, what is the length of the hypotenuse ACACAC?

Correct Answer: A

Q23.A girl of height 1.5 m walks away from a lamp post at a speed of 1 m/s. If the lamp is 4.5 m above the ground, what will be the length of her shadow after 5 seconds?

Correct Answer: B

Q24.In triangle ABC, if DE is parallel to BC and AD = 1.5 cm, DB = 3 cm, and DE = 1 cm, find the length of EC.

Correct Answer: A

Q25.In triangle △ABC\triangle ABC△ABC, a line DEDEDE is drawn parallel to BCBCBC. If AD=7.2AD = 7.2AD=7.2 cm, DC=5.4DC = 5.4DC=5.4 cm, and DE=1.8DE = 1.8DE=1.8 cm, what is the length of BCBCBC?

Correct Answer: D

Q26.In triangle △PQR\triangle PQR△PQR, SSS and TTT are points on PRPRPR and QRQRQR respectively such that ∠P=∠RTS\angle P = \angle RTS∠P=∠RTS. If PS=3PS = 3PS=3 cm, SR=6SR = 6SR=6 cm, QT=4QT = 4QT=4 cm, and TR=8TR = 8TR=8 cm, what is the length of STSTST?

Correct Answer: A

Q27.In a trapezium ABCDABCDABCD with AB∥CDAB \parallel CDAB∥CD and diagonals ACACAC and BDBDBD intersecting at point OOO, if △AOB∼△COD\triangle AOB \sim \triangle COD△AOB∼△COD, which of the following is true?

Correct Answer: C

Q28.In a triangle △PQR\triangle PQR△PQR, a line EFEFEF is drawn parallel to QRQRQR such that EEE is on PQPQPQ and FFF is on PRPRPR. If PE=3.9PE = 3.9PE=3.9 cm, EQ=3EQ = 3EQ=3 cm, PF=3.6PF = 3.6PF=3.6 cm, and FR=2.4FR = 2.4FR=2.4 cm, is EF∥QREF \parallel QREF∥QR?