TRIANGLE

 TRIANGLE

• Geometry, a triangle is a three-sided polygon that consists of three edges and three vertices. 
• The most important property of a triangle is that the sum of the internal angles of a triangle is equal to 180 degrees

DEFINITION

• As we discussed in the introduction, a triangle is a type of polygon, which has three sides, and the two sides are joined end to end is called the vertex of the triangle. 
• An angle is formed between two sides. This is one of the important parts of geometry.

Types

On the basis of length of the sides, triangles are classified into three categories:

1.     Scalene Triangle

2.     Isosceles Triangle

3.     Equilateral Triangle

On the basis of measurement of the angles, triangles are classified into three categories:

1.     Acute Angle Triangle

2.     Right Angle Triangle

3.     Obtuse Angle Triangle

Scalene Triangle

• A scalene triangle is a type of triangle, in which all the three sides have different side measures. 
• Due to this, the three angles are also different from each other.

Isosceles Triangle

• In an isosceles triangle, two sides have equal length. 
• The two angles opposite to the two equal sides are also equal to each other.

Equilateral Triangle

• An equilateral triangle has all three sides equal to each other. 
• Due to this all the internal angles are of equal degrees, i.e. each of the angles is 60°.

Acute Angled Triangle

• An acute triangle has all of its angles less than 90°.

Right Angled Triangle

• In a right triangle, one of the angles is equal to 90° or right angle.

Obtuse Angled Triangle

• An obtuse triangle has any of its one angles more than 90°.

Perimeter of Triangle

• A perimeter of a triangle is defined as the total length of the outer boundary of the triangle. 
•  we can say, the perimeter of the triangle is equal to the sum of all its three sides. 
• The unit of the perimeter is same as the unit of sides of the triangle. 

If ABC is a triangle, where AB, BC and AC are the lengths of its sides, then the perimeter of ABC is given by:

Perimeter = AB+BC+AC

Area of a Triangle

• The area of triangle is the region occupied by the triangle in 2d space. 
• The area for different triangles varies from each other depending on their dimensions. 
• We can calculate the area if we know the base length and the height of a triangle. 
• It is measured in square units.

Suppose a triangle with base ‘B’ and height ‘H’ is given to us, then, the area of a triangle is given by-

 

Formula:

Area of triangle =  Half of Product of Base and Height Area = 1/2 × Base × Height

PROPERTIES OF TRIANGLE

• Sum of angles of the triangle is equal to 180 degrees.
• Exterior angles of a triangle add up to 360 degrees.
• Shortest side is always opposite the smallest angle of a triangle.

Examples

1.Find the area of a triangle having base equal to 9 cm and height equal to 6 cm.

Solution -  We know that Area = 1/2 × Base × Height

= 1/2 × 9 × 6 cm2

= 27 cm2

2.In a right-angled triangle, ∆ABC, BC = 26 units and AB = 10 units. If BC is the longest side of the triangle, then what is the area of ∆ABC?

SOLUTION - We know that ∆ABC is a right-angled triangle
BC = 26 units
AB = 10 units
BC is the longest side of the triangle

According to Pythagoras rule:

BC2 = AB2 + AC2

262 = 102 + AC2

AC2 = 676 – 100 = 576

Therefore, AC = 24 units

We know that the area of a right-angled triangle = ½ * product of the two perpendicular sides = ½ * AB * AC = ½ * 10 * 24 = 120 sq. units