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In this post I will share some very important formulae for Geometry. Geometry is all about theorems and properties and there are endless things to mug in it, but I will only mention those formulae that are asked by SSC every year and hence are indispensable. Make sure you absorb all these well.

__Orthocentre, Incentre, Circumcentre and Centroid__ :

This topic is very important from SSC point of view. You all must be knowing the theory of these terms, but I will reiterate –

__Orthocentre__– Point of intersection of altitudes of a triangle

__Incentre__– Point of intersection of angle bisectors

__Circumcentre__– Point of intersection of perpendicular bisectors

__Centroid__– Point of intersection of medians

The above information, although important, is not sufficient enough to solve the questions. Each of these terms require different approach to solve the questions based on them.

__Orthocentre__

**∠AHB + ∠ACB = 180 [Whenever you see the word ‘orthocentre’ in the paper, you should immediately recall this formula]**

Over the years SSC has asked many questions based on this simple formula. Like –

**Answer : (A)**

__Incentre__

**∠AIC = 90 + ∠ABC/2**[Whenever you see the word ‘incentre’ in the paper, you should immediately recall this formula]

__Circumcentre__While solving questions on circumcentre, you should imagine the above figure. Here two things are quite useful –

∠AOB = 2∠ACB

OA = OB = OC [radius of the circle]. And hence,

∠OAB = ∠OBA

∠OCA = ∠OAC

∠OBC = ∠OCB

Each angle of an equilateral triangle is 60 and we have seen earlier that the angle at the centre is twice that of the triangle.

Hence **∠**AOC = 120

**Answer : (C)**

**Q. 3**

**Answer : (D)**

__Centroid__

Remember one more property related to centroid,

Area(ΔOBC) = 1/3Area(ΔABC)

We know DG = 1/2AG

Hence DG = 2cm

**Answer : (A)**

__Some additional things to remember :__

**1) The orthocentre, incentre, circumcentre and centroid of an equilateral traingle coincide, i.e., a single point acts as all the centres.**

**2) For an equilateral triangle-**

- Inradius = h/3 = a/2√3
- Circumradius or outer-radius= 2h/3 = a/√3
- Height(h) = (√3/2)a

where a = side of the equilateral triangle

**For a right-angled triangle**

- Orthocentre is at the right angle vertex
- Circumcentre is the midpoint of the hypotenuse

Here is a question based on this fact-

Radius of the circumcircle = 15, hence diameter = 30cm

AC = 30cm

We have to find AB and BC.

AB = BC [as ∠ACB = ∠ABC = 45]

AB = sin45 * AC = 30/√2 = 15√2

**Answer : (C)**

__Interior and Exterior Angles__

Another important topic of geometry is “Internal and External Angles” of polygon.

Now remember few formulae for such questions-

1) Sum of a interior angle and its corresponding exterior angle adds up to 180 degrees.

2) Sum of all exterior angles of a regular polygon is always 360.

3) Sum of interior angles = (n-2)*180

4) Each interior angle of a regular polygon is equal to [(n-2)*180]/n

5) Each exterior angle of a regular polygon is equal to 360/n

Mug all these formulae well and you will be able to solve all questions on interior and exterior angles easily.

Now let’s solve some CGL questions

Given

(n – 2)*180 = 2*360

So n – 2 = 4

or n = 6

**Answer : (B)**

**Note :** Here the difference between the “angles” of two polygons is given. You might be wondering that they haven’t mentioned the type of angle, whether interior or exterior. Well, it doesn’t matter! Because as far as two polygons are concerned-

Difference between their interior angles = Difference between their exterior angles

But I will take the angles to be exterior, because that will make our calculations simple

Let the sides be 5x and 4x.

Measure of each exterior angle of Polygon 1 = 360/5x = 72/x

Measure of each exterior angle of Polygon 1 = 360/4x = 90/x

Given, 90/x – 72/x = 6

so x = 3

The two angles are 5x and 4x

5x = 5*3 = 15

4x = 4*3 = 12

**Had I taken the angles to be interior – **

We have seen earlier that the measure of each angle of a polygon is [(n – 2)*180]/n.

Given [(5x – 2)*180]/5x – [(4x – 2)*180]/4x = 6

Solve it and you will get x = 3

So the angles are 15 and 12

**Answer : (D)**

**Note : Please note that each and every formula that I have mentioned in this post is extremely important and ****I can guarantee that after mugging all these formulae you will be able to solve all the questions on ****orthocentre, incentre, circumcentre, centroid and interior-exterior angles…**

**Keep reading 🙂**