eVidyarthi

Main Menu
  • eVidyarthi
  • School
    • Class 6th
      • Maths Class 6
      • Science Class 6
      • Hindi Class 6
      • व्याकरण
      • English Class 6
      • English Grammar
      • Sanskrit Class 6
      • Geography
      • Civics
      • History
    • Class 7th
      • Maths Class 7
      • Science Class 7
      • Hindi Class 7
      • व्याकरण
      • English Class 7
      • English Grammar
      • Sanskrit Class 7
      • Geography
      • Civics
      • History
    • Class 8th
      • Maths Class 8
      • Science Class 8
      • Hindi Class 8
      • व्याकरण
      • English Class 8
      • English Grammar
      • Sanskrit Class 8
      • Geography
      • Civics
      • History
    • Class 9th
      • Maths Class 9
      • Science Class 9
      • Hindi Class 9
      • English Class 9
      • English Grammar
      • व्याकरण
      • Economics Class 9
      • Geography Class 9
      • Civics Class 9
      • History Class 9
    • Class 10th
      • Maths Class 10
      • Science Class 10
      • Hindi Class 10
      • English Class 10
      • English Grammar
      • व्याकरण
      • Economics Class 10
      • History Class 10
      • Civics Class 10
      • Geography Class 10
    • Class 11th
      • Maths Class 11
      • Accounts Class 11
      • English Class 11
      • English Grammar
      • व्याकरण
      • Chemistry Class 11
      • Physics Class 11
      • Biology Class 11
    • Class 12th
      • Maths Class 12
      • Accounts Class 12
      • Chemistry Class 12
      • Physics Class 12
      • Biology Class 12
      • English Class 12
      • English Grammar
      • व्याकरण
    • Close
  • English
    • Basic English Grammar
    • Basic English Speaking
    • English Vocabulary
    • English Idioms & Phrases
    • Personality Enhancement
    • Interview Skills
    • Close
  • Softwares
    • Microsoft Word
    • PhotoShop
    • Excel
    • Illustrator
    • PowerPoint
    • Close
Class 10th Maths || Menu
  • Important
    • Previous Year Papers 2019
    • CBSE Board Paper 2018
    • CBSE Board Paper 2017
    • Sample Papers
    • Sample Paper – I
    • MCQs
    • Important Formulas
    • Revision Notes
    • Mathematics Book
    • Marking Scheme
    • Mathematics Syllabus
    • Close
  • Real Numbers
    • Ex. 1.1 & Intro
    • Ex. 1.2 NCERT
    • Ex. 1.3 & Ex. 1.4
    • R.D Sharma Solutions
    • Important Formulas
    • NCERT Solutions
    • Close
  • Polynomials
    • Ex. 2.1 & Intro
    • Ex. 2.2 NCERT
    • Ex. 2.3 & Ex. 2.4 NCERT
    • R. D Sharma Solutions
    • NCERT Solutions
    • Close
  • Pair of Linear Equations in 2 Variables
    • Ex. 3.1 & Intro
    • Ex. 3.2 NCERT
    • Ex. 3.3 NCERT
    • Ex 3.4 NCERT
    • Ex 3.5 NCERT
    • Ex. 3.6 & Ex. 3.7 NCERT
    • R. D Sharma Solutions
    • NCERT Solutions
    • Close
  • Quadratic Equations
    • Ex 4.1 & Intro
    • Ex 4.2 NCERT
    • Ex. 4.3 NCERT
    • Ex. 4.4 NCERT
    • R. D Sharma Solutions
    • NCERT Solutions
    • Close
  • Arithmetic Progression
    • Ex 5.1 & Intro
    • Ex 5.2 NCERT
    • Ex 5.3 NCERT
    • Ex 5.4 NCERT
    • R. D Sharma Solutions
    • NCERT Solutions
    • Close
  • Triangles
    • Intro & Theorem
    • Ex 6.2 NCERT
    • Ex. 6.3 NCERT
    • Ex 6.4 NCERT
    • Ex 6.5 NCERT
    • Ex. 6.6 NCERT
    • R D Sharma Solutions
    • NCERT Solutions
    • Close
  • Coordinate Geometry
    • Ex. 7.1 & Intro
    • Ex. 7.2 NCERT
    • Ex. 7.3 NCERT
    • Ex 7.4 NCERT
    • R D Sharma Solutions
    • NCERT Solutions
    • Close
  • Introduction to Trigonometry
    • Ex 8.1 & Intro
    • Ex. 8.2 NCERT
    • Ex 8.3 NCERT
    • Ex 8.4 NCERT
    • Ex. 6.1 with Examples – R.D Sharma
    • R D Sharma Solutions
    • NCERT Solutions
    • Close
  • Some Applications of Trigonometry
    • Ex. 9.1 & Intro
    • R D Sharma Solutions
    • NCERT Solutions
    • Close
  • Circles
    • Intro & Theorem
    • Ex. 10.1 & Ex. 10.2 NCERT
    • R D Sharma Solutions
    • NCERT Solutions
    • Close
  • Constructions
    • Construction of Similar Figures
    • Construction of Tangents
    • NCERT Solutions
    • Close
  • Areas Related to Circles
    • Ex 12.1 & Intro
    • Ex 12.2 NCERT
    • Ex. 12.3 NCERT
    • R D Sharma Solutions
    • Important Formulas
    • NCERT Solutions
    • Close
  • Surface Areas and Volumes
    • Ex. 13.1 NCERT
    • Ex. 13.2 NCERT
    • Ex. 13.3 NCERT
    • Ex. 13.4 & Ex. 13.5 NCERT
    • R. D Sharma Solutions
    • NCERT Solutions
    • Close
  • Statistics
    • Ex. 14.1 NCERT
    • Ex. 14.2 NCERT
    • Ex. 14.3 NCERT
    • Ex. 14.4 NCERT
    • R D Sharma Solutions
    • NCERT Solutions
    • Close
  • Probability
    • Ex. 15.1 & Intro
    • Ex. 15.2 NCERT
    • R D Sharma Solutions
    • NCERT Solutions
    • Close

CBSE Class 10 Maths Chapter 6 Triangles Notes

SIMILAR FIGURES

  • Two figures having the same shape but not necessary the same size are called similar figures.
  • All congruent figures are similar but all similar figures are not congruent.

SIMILAR POLYGONS
Two polygons are said to be similar to each other, if:
(i) their corresponding angles are equal, and
(ii) the lengths of their corresponding sides are proportional

Example:
Any two line segments are similar since length are proportional
Triangles Class 10 Notes Maths Chapter 6 1

Any two circles are similar since radii are proportional
Triangles Class 10 Notes Maths Chapter 6 2
Any two squares are similar since corresponding angles are equal and lengths are proportional.
Triangles Class 10 Notes Maths Chapter 6 3
Note:
Similar figures are congruent if there is one to one correspondence between the figures.
∴ From above we deduce:

Any two triangles are similar, if their
Triangles Class 10 Notes Maths Chapter 6 4

(i) Corresponding angles are equal
∠A = ∠P
∠B = ∠Q
∠C = ∠R

(ii) Corresponding sides are proportional
\frac { AB }{ PQ } =\frac { AC }{ PR } =\frac { BC }{ QR }

THALES THEOREM OR BASIC PROPORTIONALITY THEORY

Theorem 1:
State and prove Thales’ Theorem.
Statement:
If a line is drawn parallel to one side of a triangle to intersect the other two sides in distinct points, the other two sides are divided in the same ratio.
Triangles Class 10 Notes Maths Chapter 6 5
Given: In ∆ABC, DE || BC.
To prove: \frac { AD }{ DB } =\frac { AE }{ EC }
Const.: Draw EM ⊥ AD and DN ⊥ AE. Join B to E and C to D.
Proof: In ∆ADE and ∆BDE,
\frac { ar(\Delta ADE) }{ ar(\Delta BDE) } =\frac { \frac { 1 }{ 2 } \times AD\times EM }{ \frac { 1 }{ 2 } \times DB\times EM } =\frac { AD }{ DB }  ……..(i) [Area of ∆ = \frac { 1 }{ 2 } x base x corresponding altitude
In ∆ADE and ∆CDE,
\frac { ar(\Delta ADE) }{ ar(\Delta CDE) } =\frac { \frac { 1 }{ 2 } \times AE\times DN }{ \frac { 1 }{ 2 } \times EC\times DN } =\frac { AE }{ EC }
∵ DE || BC …[Given
∴ ar(∆BDE) = ar(∆CDE)
…[∵ As on the same base and between the same parallel sides are equal in area
From (i), (ii) and (iii),
\frac { AD }{ DB } =\frac { AE }{ EC }

CRITERION FOR SIMILARITY OF TRIANGLES

Two triangles are similar if either of the following three criterion’s are satisfied:

  • AAA similarity Criterion. If two triangles are equiangular, then they are similar.
  • Corollary(AA similarity). If two angles of one triangle are respectively equal to two angles of another triangle, then the two triangles are similar.
  • SSS Similarity Criterion. If the corresponding sides of two triangles are proportional, then they are similar.
  • SAS Similarity Criterion. If in two triangles, one pair of corresponding sides are proportional and the included angles are equal, then the two triangles are similar.

Results in Similar Triangles based on Similarity Criterion:

    1. Ratio of corresponding sides = Ratio of corresponding perimeters
    2. Ratio of corresponding sides = Ratio of corresponding medians
  1. Ratio of corresponding sides = Ratio of corresponding altitudes
  2. Ratio of corresponding sides = Ratio of corresponding angle bisector segments.

AREA OF SIMILAR TRIANGLES

Theorem 2.
The ratio of the areas of two similar triangles is equal to the square of the ratio of their corresponding sides.
Given: ∆ABC ~ ∆DEF
To prove: \frac { ar(\Delta ABC) }{ ar(\Delta DEF) } =\frac { { AB }^{ 2 } }{ { DE }^{ 2 } } =\frac { { BC }^{ 2 } }{ { EF }^{ 2 } } =\frac { { AC }^{ 2 } }{ { DF }^{ 2 } }
Const.: Draw AM ⊥ BC and DN ⊥ EF.
Proof: In ∆ABC and ∆DEF
Triangles Class 10 Notes Maths Chapter 6 6
\frac { ar(\Delta ABC) }{ ar(\Delta DEF) } =\frac { \frac { 1 }{ 2 } \times BC\times AM }{ \frac { 1 }{ 2 } \times EF\times DN } =\frac { BC }{ EF } .\frac { AM }{ DN }  …(i) ……[Area of ∆ = \frac { 1 }{ 2 } x base x corresponding altitude
∵ ∆ABC ~ ∆DEF
∴ \frac { AB }{ DE } =\frac { BC }{ EF }  …..(ii) …[Sides are proportional
∠B = ∠E ……..[∵ ∆ABC ~ ∆DEF
∠M = ∠N …..[each 90°
∴ ∆ABM ~ ∆DEN …………[AA similarity
∴ \frac { AB }{ DE } =\frac { AM }{ DN }  …..(iii) …[Sides are proportional
From (ii) and (iii), we have: \frac { BC }{ EF } =\frac { AM }{ DN }  …(iv)

From (i) and (iv), we have: \frac { ar(\Delta ABC) }{ ar(\Delta DEF) } =\frac { BC }{ EF } .\frac { BC }{ EF } =\frac { { BC }^{ 2 } }{ { EF }^{ 2 } }
Similarly, we can prove that
\frac { ar(\Delta ABC) }{ ar(\Delta DEF) } =\frac { { AB }^{ 2 } }{ { DE }^{ 2 } } =\frac { AC^{ 2 } }{ DF^{ 2 } }
∴\frac { ar(\Delta ABC) }{ ar(\Delta DEF) } =\frac { { AB }^{ 2 } }{ { DE }^{ 2 } } =\frac { { BC }^{ 2 } }{ { EF }^{ 2 } } =\frac { AC^{ 2 } }{ DF^{ 2 } }

Results based on Area Theorem:

  1. Ratio of areas of two similar triangles = Ratio of squares of corresponding altitudes
  2. Ratio of areas of two similar triangles = Ratio of squares of corresponding medians
  3. Ratio of areas of two similar triangles = Ratio of squares of corresponding angle bisector segments.

Note:
If the areas of two similar triangles are equal, the triangles are congruent.

 

PYTHAGORAS THEOREM

Theorem 3:
State and prove Pythagoras’ Theorem.
Statement:
Prove that, in a right triangle, the square of the hypotenuse is equal to the sum of the squares of the other two sides.
Given: ∆ABC is a right triangle right-angled at B.
To prove: AB² + BC² = AC²
Const.: Draw BD ⊥ AC
Proof: In ∆s ABC and ADB,
Triangles Class 10 Notes Maths Chapter 6 7
∠A = ∠A …[common
∠ABC = ∠ADB …[each 90°
∴ ∆ABC ~ ∆ADB …[AA Similarity
∴ \frac { AB }{ AD } =\frac { AC }{ AB }  ………[sides are proportional]
⇒ AB² = AC.AD
Now in ∆ABC and ∆BDC
∠C = ∠C …..[common]
∠ABC = ∠BDC ….[each 90°]
∴ ∆ABC ~ ∆BDC …..[AA similarity]
∴ \frac { BC }{ DC } =\frac { AC }{ BC }  ……..[sides are proportional]
BC² = AC.DC …(ii)
On adding (i) and (ii), we get
AB² + BC² = ACAD + AC.DC
⇒ AB² + BC² = AC.(AD + DC)
AB² + BC² = AC.AC
∴AB² + BC² = AC²

CONVERSE OF PYTHAGORAS THEOREM

Theorem 4:
State and prove the converse of Pythagoras’ Theorem.
Statement:
Prove that, in a triangle, if square of one side is equal to the sum of the squares of the other two sides, then the angle opposite the first side is a right angle.
Triangles Class 10 Notes Maths Chapter 6 8
Given: In ∆ABC, AB² + BC² = AC²
To prove: ∠ABC = 90°
Const.: Draw a right angled ∆DEF in which DE = AB and EF = BC
Proof: In ∆ABC,
AB² + BC² = AC² …(i) [given]
In rt. ∆DEF
DE² + EF² = DF² …[by pythagoras theorem]
AB² + BC² = DF² …..(ii) …[DE = AB, EF = BC]
From (i) and (ii), we get
AC² = DF²
⇒ AC = DF
Now, DE = AB …[by cont]
EF = BC …[by cont]
DF = AC …….[proved above]
∴ ∆DEF ≅ ∆ABC ……[sss congruence]
∴ ∠DEF = ∠ABC …..[CPCT]
∠DEF = 90° …[by cont]
∴ ∠ABC = 90°

Results based on Pythagoras’ Theorem:
(i) Result on obtuse Triangles.
If ∆ABC is an obtuse angled triangle, obtuse angled at B,
If AD ⊥ CB, then
AC² = AB² + BC² + 2 BC.BD
Triangles Class 10 Notes Maths Chapter 6 9

(ii) Result on Acute Triangles.
If ∆ABC is an acute angled triangle, acute angled at B, and AD ⊥ BC, then
AC² = AB² + BC² – 2 BD.BC.
Triangles Class 10 Notes Maths Chapter 6 10

Android App

eVidyarthi

Search On eVidyarthi

Evidyarthi on Facebook

Like us on Facebook

Follow Evidyarthi on Youtube

Learn English
Learn English Through हिन्दी
Job Interview Skills
English Grammar
हिंदी व्याकरण - Vyakaran
Mathematics Class 6th
Science Class 6th
हिन्दी Class 6th
Mathematics Class 7th
Science Class 7th
हिन्दी Class 7th
हिन्दी Class 8th
Mathematics Class 8th
Mathematics Class 9th
English Class 9th
Science Class 9th
Mathematics Class 10th
English Class 10th
Mathematics Class XI
Chemistry Class XI
Accountancy Class 11th
Accountancy Class 12th
Mathematics Class 12th
Microsoft Word
Microsoft Excel
Microsoft PowerPoint
Adobe PhotoShop
Adobe Illustrator
Learn German
Learn French
IIT JEE
Privacy Policies, Contact Us
Copyright © 2020 eVidyarthi and its licensors. All Rights Reserved.