Session 4: Triangles – Definition, Types, Interactives and Examples

What is a Triangle?

A triangle is any closed figure made by three non-parallel line segments.

If A, B and C are three non-collinear points in a plane, then the figure made up by the three line segments AB, BC and CA is called a triangle. We can name the triangle as ‘Triangle ABC’. It is common to use the symbol ‘△’ in place of the word ‘triangle’—for example, △ABC.

Parts of a Triangle

A triangle has six parts or elements, namely:

• Three sides: AB, BC and CA 
• Three angles: ∠A, ∠B and ∠C  

The point of intersection of two adjacent sides of a triangle is called a vertex. A △ABC has three vertices A, B and C. The vertex A is opposite to side BC, B is opposite to side CA, and C is opposite to side AB.

 

Position of a Point with respect to a Triangle

In the triangle below, A, B, and C are the three points. 

• The region consisting of all points lying outside a triangle is called the exterior of the triangle. Point A is an exterior point. 
• The region consisting of all points lying inside a triangle is called the interior of the triangle. Point B is an interior point. 
• The region consisting of all points which are on the triangle is called the boundary of the triangle. Point C is on the boundary of the triangle. 

The region consisting of all points which are either on the triangle or lie inside the triangle is called the triangular region. 

 

What is the Median of a Triangle?

The median of a triangle is a line segment that connects the midpoint of one side of the triangle with the opposite vertex.

Something interesting happens when you consider the three medians of a triangle.

What is the Altitude of a Triangle?

The altitude of a triangle is a line that is perpendicular to one side of the triangle and passes through the opposite vertex. The altitude is always the height of the triangle and, interestingly, sometimes occurs outside the triangle entirely.

Consider what happens when the three altitudes of a triangle intersect.

Types of Triangles

Triangles can be classified by their sides and by their angles.

Types of Triangles Based on Sides

When classifying a triangle by its sides, you should see if any sides are the same length.

• If no sides are the same length, it is a scalene triangle.
• If two sides are the same length, it is an isosceles triangle.
• If all three sides are the same length, it is an equilateral triangle.

Types of Triangles Based on Angles

When classifying a triangle by its angles, you should look at the size of the angles:

• If a triangle has a right angle, it is a right-angled triangle.
• If the measures of all angles in a triangle are less than 90∘, it is an acute triangle.
• A special case of an acute triangle is when all three angles are equal. In that case, all three angles are 60 degrees, and they form an equiangular triangle.
• If the measure of one angle in a triangle is greater than 90∘, then it is an obtuse triangle.

DID YOU KNOW?Euclid first defined an isosceles triangle to have exactly two equal sides. However, nowadays, an isosceles triangle is defined to have at least two equal sides. Hence, an equilateral triangle is a special type of isosceles triangle.

Perimeter of a Triangle

The sum of the lengths of the sides of a triangle is called its perimeter. 

In the figure below, the triangle’s three sides are represented as AB=c, AC=b, and CB=A, respectively.

PerimeterTriangle=(A+B+C) units

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Triangles – Examples

Example 1

Identify the triangle based on the length of the sides and angles given.

Remember this!

• A triangle is any closed figure made by three non-parallel line segments. 
• The three sides and three angles of a triangle are called the six parts or elements of the triangle. 
• If no sides are the same length, it is a scalene triangle. 
• If two sides are the same length, it is an isosceles triangle. 
• If all three sides are the same length, it is an equilateral triangle. 
• If a triangle has a right angle, it is a right-angled triangle. 
• If the measures of all angles in a triangle are less than 90∘, it is an acute triangle.
• If the measure of one of the angles in a triangle is greater than 90∘, it is an obtuse triangle.
•  PerimeterTriangle=(a+b+c) units