# Study Notes on Triangles

By Gaurav Mohanty|Updated : December 10th, 2021

In continuation of the IPMAT 2022 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.

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 the " 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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