Triangles

A triangle is a polygon having three sides. 
Area: 
1. Area = 1/2 x base x height 
2. Area = √s(s-a)(s-b)(s-c) where s = a+b+c/2 
3. Area = rs (where r is in-radius) 
4. Area = 1/2 x product of two sides x sine of angle 
5. Area = abc/4R where R = circumradius 

Congruency of Triangles: 
1. SAS congruency: If two sides and an included angle of one triangle are equal to two sides and an included angle of another, the two triangles are congruent. 
2. ASA congruency: If two angles and the included side of one triangle is equal to two angles and the included side of another, the triangles are congruent. 
3. AAS congruency: If two angles and side opposite to one of the angles is equal to the corresponding angles and sides of another triangle, the triangles are congruent. 
4. SSS congruency: If three sides of one triangle are equal to three sides of another triangle, the two triangles are congruent. 
5. SSA congruency: If two sides and the angle opposite the greater side of one triangle are equal to the two sides and the angle opposite to the greater side of another triangle, then triangle are congruent. 

Similarity of Triangles: 
1. AAA similarity: If in two triangles, corresponding angles are equal, then the triangles are similar. 
2. SSS similarity: If the corresponding sides of two triangles are proportional then they are similar. 
3. SAS similarity: If in two triangles, one pair of corresponding sides are proportional and the included angles are equal then the two triangles are similar. 

Equilateral triangles: 
 

1. Height = a√3/2 
2. Area = √3a²/4 
3. R(circum radius) = 2h/3 = a/√3 
4. r(in radius) = h/3 = a/2√3 
5. In equilateral triangle orthocenter, in-centre, circumcenter and centroid coincide. 

Isosceles triangle: 
 

Area = b/4√(4a² – b²) 
where b=base and a=equal sides 

Important terms: 
1. Median: A line joining the mid-point of a side of a triangle to the opposite vertex is called a radian. 
 

• A median divides a triangle in two parts of equal area. 
   
• The point where three medians meet is called centroid of the triangle. 
   
• The centroid of a triangle divides each median in ratio 2:1. 
   

2. Altitude: A perpendicular drawn from any vertex to the opposite side is called the altitude. 
 

• The point where all altitudes meet at a point is called the orthocenter of triangle. 
   

3. Perpendicular bisector: A line that is a perpendicular to a side and bisects it is the perpendicular bisector of the side. 
 

• The point at which perpendicular bisectors of the sides meet is called the circumcenter. 
   
• The circumcenter is the centre of the circle that circumscribes the triangle. 
   

4. Incentre: 
 

• The lines bisecting the interior angles of a triangle are the angle bisectors of that triangle. 
   
• The angle bisectors meet at a point called the incentre. 
   
• The angle formed by any side at the incentre is always 90° more than the half of angle opposite to the side.