# Triangles

Triangle as 2-D shape

A two-dimensional object is a flat-surfaced object having length and width but no depth. 2D shape or two-dimensional shape can be drawn on a paper.  A triangle is a three-sided closed 2D shape having three vertices as well as three angles. It is also important to know that the sum of all the interior angles of a triangle is always 180 degrees.

## Classification of Triangles on the basis of sides:

1. Scalene Triangle- A triangle in which all the sides are of different length is called Scalene Triangle. In the adjoining figure, QPR is a scalene Triangle because the side PQ is not equal to QR which is not equal to RP.

2. Isosceles Triangle - A triangle having two sides of equal length is called Isosceles triangle. In an isosceles triangle, the angles opposite to the equal sides are equal.

In the triangle ABC, the sides AB is equal to AC. Therefore, angle B is equal to angle C.

1. Equilateral Triangle - A triangle whose all the three sides are of equal length is called an equilateral triangle. The measure of each angle of a triangle is 60 degrees.

The triangle ABC is an equilateral triangle because the sides AB, BC and CA are equal with angle A, B and C as 60 degrees.

## Classification of Triangles on the Basis of Angles:

1. Acute Triangle- A triangle in which every angle measures more than zero degrees but less than 90 degrees is called acute-angled Triangle.

2. Obtuse Triangle- A triangle in which one of the angles measures more than 90 degrees but less than 180 degrees is called an obtuse-angled triangle.

3. Right Triangle - A triangle in which one of the measures of the angles exactly 90 degrees is called a right triangle.

Terms related to Triangles:

Median: A line segment joining a vertex to the midpoint of the opposite side of a triangle is called a median of a triangle. In figure ABC, D is the midpoint of AB. Thus AD forms the median of the triangle ABC.

Similarly, a median can be drawn from the midpoint of BC as well as CA. In other words, a triangle can have three medians.

Centroid: The point of intersection of the three medians of a triangle is called the centroid.

Here, the midpoint of the medians AD, BE and FC is the centroid of the triangle ABC.

Altitude: The length of the perpendicular from a vertex to the opposite side of a triangle is called its altitude, and the side on which the perpendicular is drawn is called its base.

In the triangle ABC, the perpendicular drawn to BC, that is AL is the altitude. The side BC is called the base of the triangle.

Orthocentre: The point of intersection (or concurrence) of the three altitudes of a triangle is called its orthocentre.

The meeting point (H) of the altitudes AL, CN and BM of the triangle is called the orthocentre.

Incentre and Incircle: The point of intersection of internal bisectors of the angle of a triangle is called incentre. Here, the point I which is the meeting point of the bisectors of the angles A, B and C is called Incentre.

The incentre of a circle is also the centre of the circle which touches all the sides of the triangle.

Circumcentre and Circumcircle: The point of intersection of the perpendicular bisectors of the sides of a triangle ABC is called its circumcentre.

In the figure, the perpendicular bisectors of sides AB, BC and CA of the triangle ABC intersects at point O. The point O is called the circumcentre of the triangle.

Circumcircle is the circle drawn keeping the circumcentre of the triangle as the centre such that the circle passes through all the vertices of the triangle

## What is the Perimeter of a Triangle?

The perimeter of a triangle is the sum of the length of its 3 sides.

Perimeter of triangle (P) = Side1 + Side2  +Side3

Here a, b and c are the 3 sides of the triangle.

Thus, the perimeter of triangle (P) = a+b+c

Example: What is the perimeter of a triangle with sides of 3 inches, 4 inches and 5 inches?

Solution: Perimeter of triangle = Sum of all sides

= (3 + 4 + 5) inches

= 12 inches

Therefore, the answer is 12 inches.

This principle is exactly the same for all triangles.

## Calculation of perimeter of a triangle:

### On the basis of sides, triangles are divided into three types:

1. Isosceles Triangle:  Two sides of the isosceles triangle are equal.

So, the perimeter of an Isosceles triangle = 2a + c

Example: In triangle ABC, 2 sides of the triangle (a) = 20 cm

Base (c) = 8 cm

Perimeter = 2(20) + 8

= 48 cm

1. Equilateral Triangle: All three sides of an equilateral are equal.

So the perimeter = 3X

Where x is the length of each side of the triangle.

Example: Find the perimeter of an equilateral triangle of sides 2cm.

The perimeter is 3 $\times$2 cm = 6 cm

1. Scalene Triangle: All sides of a scalene triangle are unequal.

Perimeter = s +l + m

### Area of a Triangle

Base of a triangle can be any selected side of a triangle, usually the bottom of a triangle is taken as base. For isosceles triangles the unequal side of the triangle is taken as base. Even for the right angle triangle one of the sides containing 90degree angle can be taken as base. Height of a triangle is the straight line drawn from the top vertex (opp to base) to the base of the triangle, such that the line touches the base perpendicularly making an angle 90 degree.

If the base and height of a triangle is given then the area of triangle can be calculated by the given formula:

If the sides of the isosceles or equilateral triangle are given then Pythagoras theorem can be used to find the height of the triangle. In case of a right-angled triangle also, height can be found out by using Pythagoras theorem if the length of two sides is given.