In continuation of the CAT 2021 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)2
In 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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