Concepts of Triangle

Concepts of Triangles are explained clearly below with definitions, different types of triangles and triangle properties.

 

Triangle Definition

A closed polygon (figure) with three line segments is called a triangle. A triangle consists of three sides (AB, BC, and CA) and 3 angles (∠A, ∠B and ∠C).

There are 3 types of triangles based on sides and 3 types of triangles based on angles.

Types of Triangles Based on Sides in Triangle Concepts

Equilateral Triangle

For an equilateral triangle length of all three sides of the triangle are equal.
As all the sides are equal, all the angles are also equal for an equilateral triangle.

 

Isosceles Triangle

For an isosceles triangle, the lengths of the two sides are equal. Two angles opposite these two sides are also equal.

 

Scalene Triangle

For a scalene triangle, all the three sides’ lengths are unequal or have different lengths. As all the three lengths are unequal similarly all three angles also unequal.

Types of Triangles Based on Angles in Triangle Concepts

Acute-Angled Triangle

In an acute-angled triangle, all the angles are less than 90°, or simply we can say all the angles are acute.

 

Right-Angled Triangle

In a right-angled triangle one is equal to 90° or simply we can say one angle is a right angle.

 

Obtuse-Angled Triangle

In an obtuse-angled triangle one angle is obtuse or simply we can say one angle is greater than 90°.

Similarity and Congruency in Triangles Concepts

Similar Triangles

When two triangles corresponding angles are equal and if length of corresponding sides are proportional then those two triangles are said to be similar triangles.

It is denoted by    ΔABC ~ ΔDEF
Read as triangle ABC is similar to triangle DEF.
∠A= ∠E, ∠B= ∠D,  &  ∠C= ∠F and
(AB/DE)=(BC/DF)=(AC/EF)

Similarity Conditions in Triangle Concepts

 

1. SIDE-SIDE-SIDE (SSS) SIMILARITY CONDITION

When three sides of a triangle are proportional to corresponding sides of another triangle then those two triangles are said to be similar by SSS-similarity condition.

Here,
ΔABC ~ ΔDEF
(AB/DE)=(AC/DF)=(BC/EF)

2. SIDE- ANGLE-SIDE (SAS) SIMILARITY CONDITION

When correspondence sides of two triangles are proportional one included angle is equal then those two are said to be similar by SAS similarity condition.

Here,
ΔABC ~ ΔDEF
∠B=∠E and (AB/EF)=(BC/DE)

3. ANGLE-ANGLE-ANGLE (AAA) SIMILARITY CONDITION:

When three correspond angle of two triangles are equal then, those two triangles are said to be similar condition.
Here,
ΔGHI ~ ΔJKL
∠G=∠J, ∠H=∠K, ∠I=∠L

Congruent Triangles in Triangle Concepts

When all the sides of a triangle are equal to corresponding side and corresponding angles of another triangle then those triangles are said to be congruent triangles (or) simply we can say it as if both the triangles are exactly same to each other is sides and angles.

Denoted by ΔABC ≅ ΔDEF

Read as  triangle ABC is congruent to triangle DEF

CONGRUENCY CONDITIONS

 SIDE-SIDE-SIDE (SSS) CONGRUENCY CONDITION

 When three sides of a triangle are equal to corresponding sides of another triangle then those triangles are said to be congruent by sss congruency condition.

ΔABC ≅ ΔDEF
AB=DE, AC=DF & BC=EF

SIDE-ANGLE-SIDE CONGRUENCY CONDITION

 When two corresponding sides of two triangles and one included angle is equal then those two triangles are said to be congruent by SAS congruency condition.

ΔABC ≅ ΔDEF
AB=DE, ∠B=∠E, BC=EF

ANGLE-SIDE-ANGLE CONGRUENCY CONDITION

 When two angles and included side of a triangle are equal to corresponding two angles and the included side of another triangle then those two triangles are said to be congruent by ASA Congruency condition.

ΔABC  ≅ ΔDEF
∠ABD=∠CBD
AB=BC, ∠BOA and ∠BDC

RIGHT-HYPOTENUSE-SIDE (RHS) CONGRUENCY CONDITION

        When two sides of a right angled triangle are equal to any two corresponding sides of another right-angled triangle then those triangles are said to be congruent by RHS congruency condition.

∠B=∠D=90°
AB=DE
AC=DF
ΔABC ≅ ΔDEF

Median of Triangle

Median of a triangle is a line segment drawn from a vertex to midpoint of the opposite side of a triangle.
In ΔABC,
AD, BE and CF are medians and
BD=DC, AE=EC, AF=FB

Centroid of a Triangle

 Centroid of a triangle is the point of intersection of three medians in a triangle (or) simply we can say the point of correspondence of a medians of a triangle.

 Note:

1. centroid divides the median in the ratio 2:1 here larger part will be towards vertex and shorter part towards the base.
   For ΔABC, O is centroid
   and (OA/OD)=(OC/OF)=(OB/OE) = (2/1)
2. The areas of triangles formed by median in a triangle are equal. 6 triangles are formed by medians.
   ar ΔAOF= ar ΔOBF= ar ΔOBD=ar ΔODC= ar ΔCOE=ar ΔEOA= (1/6)×ΔABC

Altitude of a Triangle

Altitude of a triangle is a perpendicular line segment drawn from any vertex to the opposite side of triangle. Simply we can say it as triangle height.

Note:

1. A triangle has 3 altitudes.
2. For obtuse angled triangle at least one altitude lies outside the triangle.
3. Angle made by altitude is 90°.

Orthocenter of a Triangle

Ortho center of a triangle is the point of intersection of three altitudes of a triangle. Ortho center triangle denoted by H.
H is the orthocenter of a ΔABC

Note:

1. In ΔABC
   ∠BHC + ∠BAC = 180˚
   ∠CHA + ∠CBA = 180˚
   ∠AHB + ∠ACB = 180˚
2. AD, BE, CF are altitude of ΔABC and AD ⊥ BC, BE ⊥ AC, CF⊥ AB.

Perpendicular Bisector of a Triangle

Perpendicular bisector of a triangle is a line that bisects the sides of a triangle at right angle.

Note:

1. AD⊥BE and BD=DC
   BE⊥AC and EC=EA
   CF⊥AB and AF=FB
2. AD, BE, CF are perpendicular bisectors of ΔABC.
3. Perpendicular bisectors may (or) may not pass through the vertices of a triangle.

Angle Bisector of  a Triangle

It is a line that bisects the angle of a triangle.

Note:

1. It divides the opposite side into two segments that are proportional to other two sides of the triangles.
2. Angle bisectors are two types. Internal angle bisector and external angle bisector

Incentre of a Triangle

Incenter of a triangle is the point of intersection of angle bisectors of a triangle. It is the center of its inscribed circle and is represented by “I”

• AD, BE, CF are angular bisectors.
• ID, IE & IF are in-radii.

Circumcenter of a Triangle

Circumcenter of a triangle is the point of intersection of perpendicular by sectors of a triangle. It is the center of its circumscribed circle and is represented by symbol ‘O’

Note:

1. ∠BOC = 2∠BAC
   ∠COA  = 2∠CBA
   ∠AOB = 2∠ACB
2. AD, BE, CF are perpendicular bisectors
3. OC, OA, OB are circum-radii.

Example Problems in Triangle Concepts

1. If orthocenter and the centroid of a triangle are same then the triangle is

1. Right-angled
2. Equilateral
3. scalene
4. obtuse angle

Ans. (b)

2. If orthocenter, centroid, in center and circumcenter are same then the triangle is

1. equilateral
2. scalene
3. obtuse-angled
4. right-angled

Ans. (a)

3. If in-center of an equilateral triangle lies inside the triangle and its radius is 3 cm then length of the side of equilateral triangle is

solution: –     let ABC is equilateral triangle.
Given,
in-radius =ID=3 cm.
for equilateral triangle perpendicular bisector, medium, altitude, angular bisector are same.
So, since   (AI/ID)=(2/1) ( median ratio)
AI/3 =2/1 ⇒AI= 6 cm.
Height of the equilateral triangle is AD = AI + ID⇒ 6  + 3
= 9c.m………………eq(1)
Let side of equilateral triangle is  ‘A’.
Height formula for equilateral triangle is (√3/2)×a
We have ,
Area of equilateral triangle is
(½) ×b×h  = (√3/4)×a²
½ × a×h  = (√3/4)a²
           h = (√3/2)a²……………………………………………………….eq(2)

from eq (1) & (2)
(√3/2)a = 9
a = 18/√3
=  (18*√3)/3
=  6√3 cm