TRIANGLE TUTORIAL

                                      

We all see triangular shape in our daily life. In fact, the basic geometrical shape which is found in a complex construction is triangle. The tower, bridge, pyramid etc. are the common things where we can see triangle.

In fact, triangular shapes provide more support, stability against the load than rectangular shape.
However, let's learn some basic things about basic things related to triangles.

Definition:

 A triangle is a 2D (2 dimensional) confined figure with three straight sides and three angles.

Triangle

Not triangle

Triangle is coined from two words "Tri" and  "angle". "Tri" means three.  I.e triangle is a figure with three angles.
When we work with many geometrical shape in exercise book, it is necessary to specify the various figure. So, how will we denote a triangle?

 How a triangle is named:

                                                                       

Triangle is a confined figure with 3 sides. So, there are 3 vertices ( Vertex: The joining point of two sides ; plural form: Vertices ).
 Generally, a triangle is named with 3 vertices.
                                                    

Triangle ABC

A triangle with vertices A,B,C is denoted as △ABC.

Various terms related to Triangle:

                                                      

Side:

The line segment ( i.e AB, BC, CA ) are called sides of the triangle.

Vertex:

The joining point of two sides (i.e A,B,C) are called vertex ( Plural- vertices ).

Interior ( or Internal) angle of Triangle:

                                                                       

Internal angle is the angle between two sides and placed internally.

For the adjacent figure, ∠ABC, ∠BCA, ∠CAB are the interior angel of triangle ABC.

Exterior Angle of Triangle:

If we extended a side of a triangle, the angle between the extended part and the adjacent side is known as exterior angle.

             

In the figure ∠ACD,∠ABE,∠CBG,∠BCF are exterior angle.

Types of Triangle:

Triangles can be classified on the basis of 

• The length of their side.
• Internal angle.

Types of Triangle classified on the basis of their length:

In how many types may triangle be classified? Think ..... Yes, you are right.

• Triangle whose all sides are equal in length.
• Triangle whose two sides are equal in length.
• Triangle whose all sides are different length.

On the basis of length of sides triangle are of 3 types. 

i) Equilateral Triangle:

          

Equilateral triangle

                              

           

Length of the sides of this type of triangle are equal.

Hence, all 3 internal angles are 60° each.

     

ii) Isosceles Triangle:

This type of triangle has two equal sides.
Hence, two angles of this triangle ( angle between equal side and unequal side) are equal.
                                         
         

Triangle ABC is an isosceles triangle. Its AC and BC sides are equal. Hence, ∠CAB,∠ABC are also equal.

iii) Scalene Triangle:

                                               

Scalene Triangle

The triangle whose all sides are of different length is known as Scalene Triangle.

Since, all sides are different in length, the internal angles also different in measure.

Types of Triangle classified on the basis of internal angles:

                                    

Right Triangle / Right-angled Triangle:

                                            
                      

The triangle whose one interior angles is 90° is known as Right Triangle / Right-angled Triangle.
The triangle  in adjacent diagram is a Right Triangle. Its ∠BAC is right angle i.e 90° .

Some properties of Right Triangle:                            

                                       

       

•   The side opposite to the right angle is called  Hypotenuse.
  Here, BC is hypotenuse.

•  Two another sides (except hypotenuse ) are known as Leg.
  Here, AC and AB are Leg.   

•  Hypotenuse is longest side  of a right-angled Triangle.
   
• A triangle can have only one right angle ( since, total interior angle of triangle is 180° )
   
• A right-angled triangle can be isosceles or scalene but it can't be equilateral. 
  
  Why ? .... Recall the triangle type divided on the basis of length.  All interior angle of equilateral triangle is 60°.
• The sum of the square of the length of two legs is equal to the square of the length of hypotenuse.
   This theorem is known as Pythagorean Theorem.
   So, if we apply this on the triangle just shown above it would be  as 
     
    (AC)²+(AB)²=(BC)²

Oblique Triangle:    

Triangle that do not have right angle is known as Oblique Triangle.

Acute Triangle:

Triangle whose all internal angle are acute angle is known as Acute Triangle.

** Angle less than 90° is called acute angle.

Acute Triangle

Some properties of Acute Triangle:

• All internal angles are acute angle.
   
• It can be equilateral , isosceles or scalene triangle.
   
• If a acute triangle have sides of length a,b,c  and c is the longest side, then,
  a²+b²>c²    

Do you want to know why are the relation between the sides such that?  If, Yes , read the following otherwise, skip the following paragraph.

Recall the relation associated with right-angled triangle .
                 

Look the right triangle ABC. Suppose, we want to keep same the length AB and BC but we want to  reduce the angle ∠ABC (now it is 90°).
Think ....what would be happened ? .....Yes, ... the length AB will be reduced. Now, let the new triangle would be like DEF.
where, AB=DE=3cm; BC=DE=4cm; AB=5cm BUT EF=4.4cm.
If we apply Pythagorean Theorem on triangle ABC ,then
         (AB)²+(BC)² = (AC)²
Or, 3²+4² = 5²
So, we can write 3²+4² > 4.4² 

Hence,(DE)²+(DF)² > (EF)²

Obtuse Triangle:

The triangle whose one of the internal angles is Obtuse angle is called Obtuse triangle.

** Angle more than 90° but less than 180° is known as Obtuse angle.

Some property of Obtuse Triangle:

• A obtuse triangle can have only one obtuse angle.
• It can be isosceles triangle or scalene triangle.
• If, a,b,c are the length of sides of a obtuse triangle and c is the longest side then,
     a² + b² < c²

Do you want to know why?   See the Property of Acute Triangle. I have already explained for acute angle. I think, if you understand that, you will understand it automatically.


You have read the types of triangle and some basic term associated with triangle so far. Now, I discuss the condition of existence of a triangle.
                                 

Hanger

  Suppose, you want to make a hanger. Hence, you take 3 pieces of rod of length 30cm, 15cm 7 cm.
In fact, you can not make a triangle of sides 30 cm, 15cm, 7cm. All right.
Let's know the condition of existence of a triangle.

Condition of existence of a Triangle:

Condition on Sides:

If sum of the length of any two sides is less than that of the remaining side, by using those sides, a triangle cannot be formed.
 Let, 3 sides are a,b,c.
Now a triangle can be constructed by using a,b,c if they follow the all 3 following conditions.

a+b>c;

b+c>a;

c+a>b; 

Recall the example, I gave first. The sides were 30cm, 15cm, 7cm.
Now,

30+15=45>7;

15+7=22<30;

It is evident that sides does not follow the necessary conditions. Hence, we can't construct a triangle using these length. 

Why is it so?

                    

AB is a rod and ACB is a rubber band. End points of rubber band are stuck at the point A and B. Now, AC+CB= AB
If we want to give a shape of triangle, the point C is to be kept away.
                               

If we do this, the rubber band have to be stretched. The above figure shows the structure. what does it imply?
 It implies that as the rubber band is stretched, the latter rubber band is longer than the former. i.e
          AC+CB>AB (for the latter).
So, The sum of two sides be greater than the remaining side of a triangle.

Condition on the angles:

The given angles can form a triangle if and only if both conditions hold

1. all angles be positive
2. the sum of three angles is equal to 180°.

In our real life we see many triangular place. If we want to measure the area, how shall we do so?

 Area of a Triangle:

The area of a triangle refer the area bounded by three sides of the triangle.
                                   

If we tell about the area of the △ABC, it refers the lined area of the triangle ABC. In our real life, sometimes area of the triangle is to be found. However, now we learn how to find area of a triangle. There are many formula to find this. here, I discuss two really important and useful formula only.

Process I:

When all 3 sides of a triangle are known, we can find the are of a triangle using this process. 
 Let, length of 3 sides of a triangle is a,b,c.
Therefore,  Semiperimeter= S=(a+b+c)/2
and Area = √(s(s-a)(s-b)(s-c))
 This formula is known as Heron's Formula.

Example:
Let, a triangle of sides 5cm,6cm,7cm. We have to find the area of the triangle.
Answer: 
Here sides are 5cm,6cm,7cm

 ∴  Semiperimeter= (5+6+7)/2 cm

=18/2 cm

=9cm

 Area=√(9(9-5)(9-6)(9-7))cm²
= √(9 × 4 × 3 ×2)
= √ (3 × 3 × 2 × 2 × 3 × 2) cm²
=6 √6 cm²

NOTE:

• If all 3 sides are known, we can only use this formula.
• If the length of one side of equilateral triangle is given, we can apply this formula. As all sides of equilateral triangle are equal.
• Similarly, we can find the area of isosceles triangle if the length of two unequal sides are known.

Process II:

Before learning the Process II, we have to know What the term "height" refers to.

Height:

If a perpendicular is drawn from a vertex of a triangle on the opposite side of that triangle, the length of that perpendicular known as height.
 The side on which perpendicular is drawn is known as base corresponding to that height.
             

                 

      For the above triangle ABC, AD is the height and BC is base.

Let's learn the process.
If we know the base and height of a a triangle,

The area = (1/2) base × height 

Example:

Let, the length of base and height of a triangle are 3.5 cm and 2.7 cm respectively. We have to find the area of that triangle.

Answer:

                  Area = (1/2) × 3.5 × 2.5 cm²

=4.725  cm²

 

NOTE:

• If height and base are known, we can only use this formula.
   
• For the right-angled triangle, the side associated with right angle are height and base.

                             
             

Right Triangle

    

 i.e for the right triangle ABC, AB  and BC are height and base.