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Triangles Class 10th Notes - Free NCERT Class 10 Maths Chapter 6 Notes - Download PDF

Triangles Class 10th Notes - Free NCERT Class 10 Maths Chapter 6 Notes - Download PDF

Edited By safeer | Updated on Mar 19, 2022 11:03 AM IST

Triangles, the NCERT chapter that deals with the similarity of triangles. The NCERT Class 10 Maths chapter 6 notes covers an outline of the chapter triangles. The main topics covered triangles Class 10 notes, are properties for two triangles to be similar, how congruency of triangles is different from the similarity of triangles. Class 10 Math’s chapter 6 notes include frequently asked questions about the chapter. These topics can be downloaded from the triangles Class 10 notes pdf download button.

Also, students can refer,

Triangles Class 10 Notes:

Triangle is a polygon having three sides and three vertices are known as Triangle.

Types of Triangles

1. There are three types of Triangles on the basis of the length of the sides.

There are three types of Triangles on the basis of the length of the sides

2. Three types of Triangles on the basis of their angles.

There are three types of Triangles on the basis of angles

Pythagoras theorem: In a right-angled triangle, the square of the hypotenuse side is equal to the sum of the squares of the other two sides of the triangle.

Centroid, Incenter, and Circum Center-

1. Centroid of a Triangle

The point at which all the three medians of a triangle intersect is known as the centroid of that Triangle

. Centroid of a Triangle

2. Incenter of a Triangle

The incenter of a triangle is the point of intersection of the angle bisectors of the three angles of the triangle. It is the point from where a circle can be inscribed in the Triangle.

Incenter of a Triangle

3. Circumcenter of the Triangle

The circumcenter of the triangle is the point of intersection of the perpendicular bisectors of the three vertices of the triangle.

Circumcenter of the Triangle

Congruent vs. Similar Figures:


Congruent

Similar

Angles

When Corresponding angles are the same.

Corresponding angles are the same.

Sides

When Corresponding sides are the same.

When Corresponding sides are proportional.

Example



Explanation

Both the triangles have the same magnitude of angles and same magnitude side.

Both the triangles have the same angles but not

the same sides.

Symbols

1643970001776

1643970002142

Similarity of Triangles

Two Triangles will be similar if-

  • Corresponding angles of the two triangles are the same

  • The corresponding sides of the two triangles are in the same proportion.

Similarity of Triangles

Two Triangles ∆ABC ~ ∆DEF as-

∠A=∠D,∠B=∠E, ∠C=∠F

AB/DE=BC/EF=AC/DF

Basic Proportionality Theorem (Thales Theorem)

Thales theorem says if in a given triangle a line is drawn parallel to any of the sides of the triangle so that the other two sides intersect at some distinct point then it divides the two sides in the same ratio.

In ∆KMN, if PQ║MN intersects KM at P and KN at Q, then:

i) KP/PM=KQ/QN

ii) KM/KP=KN/KQ

iii) KM/PM=KN/QN

The converse of Basic Proportionality Theorem

It is converse of basic proportionality theorem, which says if in a given Triangle a straight line divides the two sides of the Triangle in the same ratio then that straight line is parallel to the third side of that Triangle.

In KMN, if KP/PM=KQ/QN then,

PQ||MN

Criteria For The Similarity of Triangles

Three criteria to find the similarity of two Triangles are :

1. AAA (angle-Angle-Angle) criteria of similarity

Two triangles are similar to each other when any two angles of one triangle are equal to any other two angles of the other triangle.

Thus, the two Triangles are similar.

Hence, ∆ABC ~ ∆PQR

2. SSS(Side-Side-Side) criteria of similarity

When in the two triangles, all the sides of one triangle are in the same ratio with respect to the sides of the other triangle, then their corresponding angles are equal. Then the two triangles are similar.

In KMN, if KPPM=KOQN then,

PQ||MN

Criteria For The Similarity of Triangles

Three criteria to find the similarity of two Triangles are :

1. AAA (angle-Angle-Angle) criteria of similarity

Two triangles are similar to each other when any two angles of one triangle are equal to any other two angles of the other triangle.

Thus, the two Triangles are similar.

Hence, ∆ABC ~ ∆PQR

2. SSS(Side-Side-Side) criteria of similarity

When in the two triangles, all the sides of one triangle are in the same ratio with respect to the sides of the other triangle, then their corresponding angles are equal. Then the two triangles are similar.

In ∆ABC and ∆DEF

AB/DE=BC/EF=AC/DF

So ∠A=∠D, ∠B=∠E, ∠C=∠F

Therefore, ∆ABC ~ ∆DEF

3. SAS(Side-Angle-Side) criteria of similarity

When in two triangles, two sides are in the same ratio with the two sides of the other triangle and the angle including those sides is equal then the two triangles are similar.

In ∆ABC and ∆KLM

AB/KL=BC/LM and ∠B=∠L

Hence, ∆ABC ~ ∆KLM

Areas of Similar Triangles

When the two similar triangles are given then the square of the ratio of their corresponding sides will be equal to the ratio of their area.

Areas of similar Triangles

If ∆ABC ~ ∆PQR, then

ar(ABC)/ar(PQR)=(AB/PQ)2=(BC/QR)2=(AC/PR)2

  • The similarity of two Triangles made in right angle Triangle

In a right angle triangle, if a perpendicular is drawn from the right angle to the hypotenuse of the Triangle, then the two Triangles formed would be similar to the whole Triangle.

Similarity of two Triangles made in right angle Triangle

In the above right angle CP is the vertex on the hypotenuse, thus :

∆ACP ~ ∆ACB

∆PCB ~ ∆ACB

∆PCB ~ ∆ACP

Significance of NCERT Class 10 Math’s Chapter 6 Notes

Triangles Class 10 notes will help to understand the formulas, statements, rules in detail.

This NCERT Class 10 Math’s chapter 6 notes also contains previous year’s questions and NCERT textbook pdf.

In offline mode, Class 10 Math’s chapter 6 notes pdf download can be used to prepare.

Class 10 Chapter Wise Notes

NCERT Solutions of Cass 10 Subject Wise

NCERT Class 10 Exemplar Solutions for Other Subjects:

Frequently Asked Questions (FAQs)

1. Are all the main derivations covered in the Class 10 Math’s chapter 6 notes?

No, all the main derivations are not covered in the notes for Class 10 Math’s chapter 6. This NCERT note is a brief of the main topics and equations covered in the chapter and can be used for revising the triangles .

2. What is the weightage of Class 10 Triangles notes in board examination?

Students can expect 4 to 5 marks questions from the notes for Class 10 Math’s chapter 46.

3. What is AAA similarity ?

As mentioned in NCERT notes for Class 10 Math’s chapter 6.

AAA (angle-Angle-Angle) criteria of similarity

If in two Triangles all the corresponding angles are equal then their corresponding sides are also in proportion.


Thus, the two Triangles are similar.


Hence, ∆ABC ~ ∆PQR

4. What is SAS similarity ?

SAS(Side-Angle-Side) criteria of similarity-

If in two triangles, two sides are in the same ratio with the two sides of the other triangle and the angle including those sides is equal then the two triangles are similar.

Articles

Get answers from students and experts

A block of mass 0.50 kg is moving with a speed of 2.00 ms-1 on a smooth surface. It strikes another mass of 1.00 kg and then they move together as a single body. The energy loss during the collision is

Option 1)

0.34\; J

Option 2)

0.16\; J

Option 3)

1.00\; J

Option 4)

0.67\; J

A person trying to lose weight by burning fat lifts a mass of 10 kg upto a height of 1 m 1000 times.  Assume that the potential energy lost each time he lowers the mass is dissipated.  How much fat will he use up considering the work done only when the weight is lifted up ?  Fat supplies 3.8×107 J of energy per kg which is converted to mechanical energy with a 20% efficiency rate.  Take g = 9.8 ms−2 :

Option 1)

2.45×10−3 kg

Option 2)

 6.45×10−3 kg

Option 3)

 9.89×10−3 kg

Option 4)

12.89×10−3 kg

 

An athlete in the olympic games covers a distance of 100 m in 10 s. His kinetic energy can be estimated to be in the range

Option 1)

2,000 \; J - 5,000\; J

Option 2)

200 \, \, J - 500 \, \, J

Option 3)

2\times 10^{5}J-3\times 10^{5}J

Option 4)

20,000 \, \, J - 50,000 \, \, J

A particle is projected at 600   to the horizontal with a kinetic energy K. The kinetic energy at the highest point

Option 1)

K/2\,

Option 2)

\; K\;

Option 3)

zero\;

Option 4)

K/4

In the reaction,

2Al_{(s)}+6HCL_{(aq)}\rightarrow 2Al^{3+}\, _{(aq)}+6Cl^{-}\, _{(aq)}+3H_{2(g)}

Option 1)

11.2\, L\, H_{2(g)}  at STP  is produced for every mole HCL_{(aq)}  consumed

Option 2)

6L\, HCl_{(aq)}  is consumed for ever 3L\, H_{2(g)}      produced

Option 3)

33.6 L\, H_{2(g)} is produced regardless of temperature and pressure for every mole Al that reacts

Option 4)

67.2\, L\, H_{2(g)} at STP is produced for every mole Al that reacts .

How many moles of magnesium phosphate, Mg_{3}(PO_{4})_{2} will contain 0.25 mole of oxygen atoms?

Option 1)

0.02

Option 2)

3.125 × 10-2

Option 3)

1.25 × 10-2

Option 4)

2.5 × 10-2

If we consider that 1/6, in place of 1/12, mass of carbon atom is taken to be the relative atomic mass unit, the mass of one mole of a substance will

Option 1)

decrease twice

Option 2)

increase two fold

Option 3)

remain unchanged

Option 4)

be a function of the molecular mass of the substance.

With increase of temperature, which of these changes?

Option 1)

Molality

Option 2)

Weight fraction of solute

Option 3)

Fraction of solute present in water

Option 4)

Mole fraction.

Number of atoms in 558.5 gram Fe (at. wt.of Fe = 55.85 g mol-1) is

Option 1)

twice that in 60 g carbon

Option 2)

6.023 × 1022

Option 3)

half that in 8 g He

Option 4)

558.5 × 6.023 × 1023

A pulley of radius 2 m is rotated about its axis by a force F = (20t - 5t2) newton (where t is measured in seconds) applied tangentially. If the moment of inertia of the pulley about its axis of rotation is 10 kg m2 , the number of rotations made by the pulley before its direction of motion if reversed, is

Option 1)

less than 3

Option 2)

more than 3 but less than 6

Option 3)

more than 6 but less than 9

Option 4)

more than 9

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