Triangle:
A triangle is a polygon having three sides. Sum of all the angles of a triangle = 1800
Types:-
- 0Obtuse angle triangle: Triangles with one of the angles more than 90. (Note=We cannot have more than one obtuse angle in a triangle.)
- Acute angle triangle: Triangles with all three angles less than 90.
- Right angle triangles: Triangles with one the angles equal to 90.
- Equilateral triangle: Triangle with all sides equal.All the angles in such a triangle measure 60
- Isosceles triangle: Triangle with two of its sides equal and consequently the angles opposite the equal sides are also equal.
- Scalene Triangle: Triangle with none of the sides equal to any other side.
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.
- In the case of aright ∠, the formula reduces to a^{2}=b^{2}+c^{2} Since cos 90=0.
- The exterior angle is equal to the sum of two interior angles not adjacent to it. ∠ACD=∠BCE=∠A+∠B
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(s−a)(s−b)(s−c) (Hero's Formula) −−−−−−−−−−−−−−−−−√
- Area=rs (where r is in radius)
- Area=1/2 x (Product of two sides x sine of the included angle)
=1/2 ac sin B
= 1/2 ab sin C
= 1/2 bc sin A
- Area= abc/4R (Where R=circum radius)
[1] Equilateral Triangles(of side a):
1. h=a.3
√2
√2
∴ sin60=3√2=h/side
2.Area=1/2(base) x (height) = 1/2 x a xa3√2 =3√2a2
3. R (circum radius) = 2h3=a3√
4.r (in radius) =h3=a23√
Properties:
- The incentre and circumcentre lie at a point that divides the height in the ratio 2:1.
- The circum-radius is always twice the radius.(R =2r)
- 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.
[2] Isosceles Triangle:
Area=b44a2−b2−−−−−−−√
1. In an isosceles triangle, the angles opposite to the equal sides are equal.
[3] 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)
R(circumradius)=hypotenus/2
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
bC2
⇒c2=pa
2:-ΔABC~ΔDAC
AC/BC=DC/AC
⇒AC2=DC×BC
⇒b2=qa
2:-ΔDBA~ΔDAC
DA/DB=DC/DA
⇒DA2=DB×DC
⇒AD2=pq
DA/DB=DC/DA
⇒DA2=DB×DC
⇒AD2=pq