# Triangles Basics | cbse24.com

## Triangle:

A triangle is a polygon having three sides. Sum of all the angles of a triangle = 1800

### Properties:

• The Sum of any two sides of a triangle has to be always greater than the third side.
• The difference between the lengths of any two sides of a triangle has to be always lesser than the third side.
• The side opposite to the greater angle will be the greater and the side opposite to the smallest angle the smallest.
• The sine rule: a/sinA=b/sinB=c/sinc=2R (where R=circum radius)
• The cosine rule:  a^{2}=b^{2}+c^{2}-2bc.cosA . This is true for all sides and respective angles.

### Area:

• Area=1/2 base x height or 1/2 b.h  (Height =Perpendicular distance between the base and vertex opposite to it)
• Area=s(sa)(sb)(sc)      (Hero's Formula)
Where, s =(a+b+c)/2    (a,b,c is the length of the sides)

### = 1/2 ab sin C

= 1/2 bc sin A

###  Equilateral Triangles(of side a):

1.  h=a.3

2
∴ sin60=32=h/side

2.Area=1/2(base) x (height) = 1/2 x a xa32 =32a2

2=osA

### Properties:

• The incentre and circumcentre lie at a point that divides the height in the ratio 2:1.
• Among all the triangles that can be formed with a given perimeter, the equilateral triangle will have the maximum area.
• An equilateral triangle in a circle will have the maximum area compared to other triangles inside the same circle.

 Isosceles Triangle:

Area=b44a2b2

1. In an isosceles triangle, the angles opposite to the equal sides are equal.

 Right-Angled Triangle:

### Pythagoras Theorem:

In the case of a right-angled triangle, the square of the hypotenuse is equal to the sum of the squares of the two sides. In the figure below for triangle ABC,

a2=b2+c2

Area = 1/2 (product of perpendicular sides)
Area=rs
(Where r= in radius and s=(s+b+c)/2 where a,b and c are sides of triangle)

⇒1/2 bc=r(a+b+c)/2
⇒r=bc/(a+b+c)

In the triangle ABC,
ΔABC~ ΔDBA ~ΔDAC

We find the following identities

1:-ΔABC~ΔDAC
∴ AB/BC=DB/BA
⇒ (AB)2=DB×BC

bC
2

c2=pa

2:-ΔABC~ΔDAC
AC/BC=DC/AC
AC2=DC×BC

b2=qa

2:-ΔDBA~ΔDAC
DA/DB=DC/DA
DA2=DB×DC