Triangle & It's Important Theorems

By Sachin Awasthi|Updated : February 23rd, 2021

In continuation of the CHSL study plan, we will discuss one of the important topics from the Geometry section i.e., " Important Theorems Related to Triangle ". Questions based on these theorems are frequently asked in the exam.

 BASIC PROPORTIONALITY THEOREM 

In a triangle, a line drawn parallel to one side to intersect the other side in distinct points divides the two sides in the same ratio

 

In continuation of the CHSL study plan, we will discuss one of the important topics from the Geometry section i.e., " Important Theorems Related to Triangle ". Questions based on these theorems are frequently asked in the exam.

 BASIC PROPORTIONALITY THEOREM 

In a triangle, a line drawn parallel to one side to intersect the other side in distinct points divides the two sides in the same ratio

As per the basic proportionality theorem:

 

Mid-point Theorem

Mid-point theorem is a special case of " Basic Proportionality Theorem ".  In the above triangle, D and E are the midpoints of the sides AB and AC respectively.

Then, as per the midpoint theorem 

DE is parallel to BC, and

DE = 1/2 BC

Pythagoras Theorem

In right triangles, the square of the hypotenuse equals the sum of the squares of the other two sides

 

As per the Pythagoras theorem: AC2 = AB2 + BC2

It is important to learn the triplets that make right-angled triangles.

3, 4, 5 

5, 12, 13

7, 24, 25

8, 15, 17

9, 40, 41

11, 60, 61

12, 35, 37

16, 63, 65

20, 21, 29

These triplets satisfy the condition of AC2 = AB2  + BC2

If a, b, c denote the sides of a triangle then

  • If c2 < a2 + b2, triangle is acute-angled.
  • If c2 = a2 + b2, triangle is right angled.
  • If c2 > a2 + b2, triangle is obtuse-angled

Median: A median of a triangle is the line from a vertex to the midpoint of the opposite side. The centroid is the point at which the medians of the triangle meet. The centroid divides the medians in the ratio 2:1. The median bisects the area of the triangles.

Theorem of Apollonius

Sum of the squares of two sides of a triangle

= 2 (median) 2 + 2 (half the third side)2In the figure AD is the median, then: AB2 + AC2 =2(AD)2 + 2(BD)2

 

 

Angle Bisector Theorem

 

If AD is the bisector of angle A, 

Then, AB/AC = BD/DC

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Sachin AwasthiSachin AwasthiMember since Nov 2019
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Javahar ali

Javahar aliJun 29, 2018

Tq so much sir
Shankar Kunchikorve
Thanks for uploading such topic please upload more topics related to geometry
shweta pal

shweta palJul 28, 2020

Thanks sir
Jitesh Mishra
Great job sir its very helpful

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