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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CBSE Learning Objectives – Key Concepts & Skills You Must Know

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.

CBSE Revision Notes & Quick Summary for Last-Minute Study

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.

CBSE Exam Tips, Important Questions & Common Mistakes to Avoid

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.

CBSE Quiz & Practice Test – MCQs, True/False Questions with Solutions

Multiple Choice Questions

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

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.

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.

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.

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

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.

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.

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.

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.

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.

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.

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.

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.

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.

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}.

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}

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.

Rectangle

Rhombus

Parallelogram

Kite

Correct Answer: D

Solution:

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

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.

\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.

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.

\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}

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

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.

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.

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.

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.

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.

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

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.

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.

\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

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.

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.

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.

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.

1.5

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

\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.

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.

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.

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.

3:4

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.

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.

\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

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.

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

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.

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

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.

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.

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.

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.

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.

\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.

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.

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

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.

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.

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.

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.

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.

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.

\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}

All circles have the same shape but can have different sizes, making them similar but not necessarily congruent.

Triangles

AI Learning Assistant

CBSE Learning Objectives – Key Concepts & Skills You Must Know

Learning Objectives

CBSE Revision Notes & Quick Summary for Last-Minute Study

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

CBSE Exam Tips, Important Questions & Common Mistakes to Avoid

Common Mistakes and Exam Tips

Common Pitfalls

Tips for Success

Multiple Choice Questions

Q1.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

Correct Answer: A

Correct Answer: A

Q4.In a triangle, if one angle is equal to one angle of another triangle and the sides including these angles are in the same ratio, what can be concluded about the triangles?

Correct Answer: B

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

Correct Answer: B

Q7.In a quadrilateral ABCD, if AB = 3 cm, BC = 3 cm, CD = 1.5 cm, and DA = 1.5 cm, are quadrilaterals ABCD and PQRS similar?

Correct Answer: B

Q8.In triangle ABC, if DE is parallel to BC and AD = 4 cm, DB = 8 cm, and DE = 2 cm, what is the length of EC?

Correct Answer: A

Q9.In a quadrilateral PQRSPQRSPQRS, if PQ=3.0PQ = 3.0PQ=3.0 cm, QR=5.0QR = 5.0QR=5.0 cm, RS=4.8RS = 4.8RS=4.8 cm, and SP=4.2SP = 4.2SP=4.2 cm, what type of quadrilateral is it based on side lengths?

Correct Answer: D

Q10.Two triangles △ABC\triangle ABC△ABC and △DEF\triangle DEF△DEF are similar. If AB=6AB = 6AB=6 cm, BC=8BC = 8BC=8 cm, CA=10CA = 10CA=10 cm, and the scale factor from △ABC\triangle ABC△ABC to △DEF\triangle DEF△DEF is 1:31:31:3, what is the perimeter of △DEF\triangle DEF△DEF?

Correct Answer: C

Q11.In Fig. 6.17, if DE∥BCDE \parallel BCDE∥BC and AD=1.5AD = 1.5AD=1.5 cm, DB=3DB = 3DB=3 cm, and DE=1DE = 1DE=1 cm, what is the length of ECECEC?

Correct Answer: A

Q12.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

Q13.Which of the following statements is true for all squares?

Correct Answer: B

Q14.In triangle PQR, if PE = 3.9 cm, EQ = 3 cm, PF = 3.6 cm, and FR = 2.4 cm, is EF parallel to QR?

Correct Answer: A

Q15.In the diagram, △ABC\triangle ABC△ABC is similar to △PQR\triangle PQR△PQR with a scale factor of 2:1. If the area of △ABC\triangle ABC△ABC is 24 cm extsuperscript{2}, what is the area of △PQR\triangle PQR△PQR?

Correct Answer: A

Q16.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

Q17.Which of the following statements is true for all circles?

Correct Answer: B

Q18.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

Q19.Consider two triangles △PQR\triangle PQR△PQR and △XYZ\triangle XYZ△XYZ such that PQXY=QRYZ=RPZX=2\frac{PQ}{XY} = \frac{QR}{YZ} = \frac{RP}{ZX} = 2XYPQ​=YZQR​=ZXRP​=2. If PQ=4PQ = 4PQ=4 cm, what is the length of XYXYXY?

Correct Answer: A

Q20.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

Q21.In a triangle PQR, if a line segment EF is drawn parallel to QR such that PEEQ=PFFR\frac{PE}{EQ} = \frac{PF}{FR}EQPE​=FRPF​, what can be said about EF?

Correct Answer: B

Q22.In a trapezium ABCD, if AB is parallel to DC and diagonals AC and BD intersect at O, which of the following is true?

Correct Answer: A

Q23.A girl of height 90 cm walks away from the base of a lamp-post at a speed of 1.2 m/s. If the lamp is 3.6 m above the ground, what is the length of her shadow after 4 seconds?

Correct Answer: B

Q24.In a triangle, if the sides are in the ratio 2:3:4, which of the following could be the angles of the triangle?

Correct Answer: D

Q25.Which of the following statements is true for two similar triangles?

Correct Answer: A

Q26.If two triangles have their corresponding sides in the same ratio, what can be said about the triangles?

Correct Answer: B

Q27.In the figure, △ABC∼△DEF\triangle ABC \sim \triangle DEF△ABC∼△DEF. If AB=6AB = 6AB=6 cm, BC=8BC = 8BC=8 cm, CA=10CA = 10CA=10 cm, and DE=3DE = 3DE=3 cm, what is the length of EFEFEF?

Correct Answer: B

Q28.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

Correct Answer: B

Q30.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?

Correct Answer: A

Q31.In Fig. 6.30, if triangle ABC has sides AB = 3.8, BC = 6, and AC = 4, and triangle PQR has sides PQ = 12, QR = 7.6, and PR = 5, which similarity criterion can be used to show that the triangles are similar?

Correct Answer: B

Q32.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

Q33.In △DEF\triangle DEF△DEF, a line PQPQPQ is drawn parallel to EFEFEF such that DP=3DP = 3DP=3 cm, PE=6PE = 6PE=6 cm, and PQ=2PQ = 2PQ=2 cm. What is the length of EFEFEF?