Classification of Triangles: Definition, Types, Properties, Examples

Classification of Triangles: A triangle is a three-sided polygon with three vertices and three edges. It is one of the most fundamental geometric shapes. 

Triangles are classified based on the length of their sides and their angles. In a polygon, a side is a line segment that connects two vertices. The figure created by two rays meeting at a common terminal point or originating at a common terminal point is known as an angle. Understanding these features enables you to apply the concepts to a variety of real-life situations. Because of their rigidity and stable structure, triangles are one of the most significant shapes and are utilised in mathematics, engineering etc.

What are Triangles?

A triangle is a plane shape made up of three non-parallel line segments.

In terms of classification, a triangle has six elements: three sides and three angles. As a result, triangles are classified using these elements.

In the triangle \(X Y Z\) given below, \(X Y, Y Z\), and \(Z X\) are three sides, \(X, Y\) and \(Z\) are three vertices and \(\angle X, \angle Y\) and \(\angle Z\) are three angles.

Perimeter of a Triangle

The perimeter of a triangle is equal to the sum of the lengths of the triangle’s sides.

If the lengths of the sides of a triangle are \(a, b\) and \(c\) units, then perimeter \(=(a+b+c)\).

The perimeter of a triangle is usually written as \(2 s\), where \(s\) is the triangle’s semi-perimeter.

Thus, \(2 s=a+b+c\).

Area of a Triangle

The area of a triangle is defined as the total space filled by its three sides on a two-dimensional plane.

\({\rm{Area}}\,{\rm{of}}\,{\rm{a}}\,{\rm{triangle}}\,\,{\rm{ = }}\frac{1}{2} \times b \times h\,{\rm{sq}}\,{\rm{units}}\), where \(b=\) base and \(h=\) height.

Properties of a Triangle

1. Angle sum property of a triangle: According to the angle sum property, the sum of a triangle’s three internal angles is always \(180^\circ \).
2. Inequality property of a triangle:
a) The length of the two sides of a triangle is greater than the length of the third side.
b) The longest side is the one on the opposite side of the biggest angle.
3. Exterior angle property: The exterior angle of a triangle is always equal to the sum of the interior opposing angles.
4. Congruence property: Two triangles are congruent if all their corresponding sides and angles are equal.
5. Similarity property: If the respective angles are congruent and the corresponding sides are proportional, two triangles are similar.

Types of Triangles

Triangles can be classified based on their internal angles and based on their sides.

There are three types of triangles based on internal angles: acute, right, and obtuse-angled triangles. Triangles are grouped into three groups based on the length of their sides: Scalene, Isosceles, and Equilateral triangles.

Types of Triangles Based on Sides

Triangles are grouped into three groups based on the length of their sides: 

1. Scalene triangle
2. Isosceles triangle
3. Equilateral triangle

Scalene triangle: A scalene triangle is a triangle with varied lengths on all three sides. A scalene triangle’s angles are also different in measure.

Here, the sides \(P Q, P R\) and \(Q R\) are unequal sides of a triangle \(P Q R\).

Heron’s formula was discovered by Heron of Alexandria, who was the first to do so. It can be used to find the area of equilateral, isosceles, and scalene triangles, as well as quadrilaterals

Area of a triangle \(=\sqrt{s(s-a)(s-b)(s-c)} \,\text {sq units}\), where \(s=\frac{\text { perimeter }}{2}=\frac{a+b+c}{2} \,\text {units}\) and \(a, b\) and \(c\) are sides of a triangle.

Properties of a Scalene Triangle

1. It is made up of three sides, each of which is unequal in length.
2. It contains three angles, each of which is different in size.
3. There are no parallel or equal sides and thus no symmetry line.
4. Acute, obtuse, or right angles can be found in the inner angles of a triangle. As a result, a scalene triangle can be an obtuse, acute, or right-angled triangle.
5. The circumscribing circle’s centre will be inside the triangle in the case of an acute scalene triangle.
6. The circumcentre of an obtuse scalene triangle will be outside the triangle.

Isosceles triangle: An isosceles triangle is a triangle with two sides of equal length.

In \(\triangle X Y Z, X Y=X Z\) and \(\angle X Y Z=\angle X Z Y\).

Properties of an Isosceles Triangle

1. The legs are the two equal sides of an isosceles triangle, and the angle between them is known as the vertex angle or apex angle.
2. The base angle is the side opposite the vertex angle, and base angles are equal.
3. The perpendicular drawn from the vertex made by the two equal sides bisects the base.

Equilateral triangle: An equilateral triangle has the same length of each of its three sides.

Here, \(\triangle P Q R\) is an equilateral triangle. Thus, \(P Q=Q R=P R\) and \(\angle P Q R=\angle Q P R=\angle P R Q=60^{\circ}\).

Area of an equilateral triangle \(=\frac{\sqrt{3}}{4} a^{2} \,\text {sq units}\), where \(a\) is the side of an equilateral triangle.

Properties of an Equilateral Triangle

1. The lengths of the sides of an equilateral triangle are the same.
2. The angles of an equilateral triangle are congruent, and each equal to \(60\) degrees.
3. Because it has three sides, it is referred to as a regular polygon.
4. An equilateral triangle’s side is bisected in equal lengths by a perpendicular drawn from each vertex to the opposite side.
5. The ortho-centre and centroid are both located at the same point.
6. For an equilateral triangle, the median, angle bisector, and altitude are the same.

Study Everything About Types of Triangles

Types of Triangles Based on Angles

Triangles are grouped into three kinds based on the measure of their angles: 

1. Acute triangle
2. Obtuse triangle
3. Right triangle

Acute Triangle: An acute triangle is a three-sided polygon having three angles that are all smaller than \(90^{\circ}\). But the sum of all three interior angles is always \(180^{\circ}\).

Properties of Acute Triangle

1. All three interior angles of an acute triangle add up to \(180^{\circ}\) according to the angle sum property.
2. At the same time, a triangle cannot be a right-angled triangle and an acute-angled triangle.
3. An acute-angled triangle and an obtuse-angled triangle cannot exist in the same triangle.
4. The acute triangle’s interior angles are always less than \(90^{\circ}\) or lie between (\(0^{\circ}\) and \(90^{\circ}\)).

Obtuse triangle: An obtuse-angled triangle, also known as an obtuse triangle, is a triangle in which one of the vertex angles is greater than \(90^{\circ}\). One of the vertex angles of an obtuse-angled triangle is obtuse, whereas the other angles are acute.

Properties of the Obtuse Triangle

1. The side of a triangle opposite to the obtuse angle is the longest.
2. There can only be one obtuse angle in a triangle. We know that a triangle’s angles add up to \(180^{\circ}\).
3. In an obtuse triangle, the sum of the other two angles is always less than \(90^{\circ}\).
4. An obtuse-angled triangle’s circumcentre and orthocentre are located outside the triangle. In an obtuse triangle, the orthocentre, the point where all a triangle’s altitudes intersect, is outside.

Right angle triangle: A right-angled triangle has one of its angles that measures \(90^{\circ}\), and the remaining two angles of a right-angled triangle are acute. A right angle is defined as a \(90^{\circ}\) angle, and a right triangle is defined as a triangle with a right angle.

Properties of a Right Triangle

1. The largest angle must be \(90^{\circ}\).
2. The hypotenuse, which is the side opposite the right angle, is the longest side.
3. The Pythagoras rule can be used to measure the lengths of the sides.

Right isosceles triangle: When the measure of each acute angle in a right-angle triangle equals \(45^{\circ}\), the triangle is known as a right isosceles triangle.

Properties of a Right Isosceles Triangle

1. A right triangle with two equal length legs is known as a right isosceles triangle.
2. The greatest angle measures \(90^{\circ}\), and the remaining two acute angles are equal, and each measures \(45^{\circ}\).

Solved Examples on Classification of Triangles

Q.1. In an isosceles triangle, if the measure of each equal angle is \(40^{\circ}\), then find the measure of the third angle.
Ans: Given that the measure of each equal angle in an isosceles triangle is \(40^{\circ}\)
Let the measure of the third angle be \(Y\).

We know that the sum of the measure of interior angles of a triangle is \(180^{\circ}\).
Now, \(40^{\circ}+40^{\circ}+Y=180^{\circ}\)
\(\Rightarrow Y=180^{\circ}-80^{\circ}=100^{\circ}\)
Hence, the measure of the third angle is \(100^{\circ}\).

Q.2. If the perimeter of an isosceles triangle is \(15 \mathrm{~cm}\), the measure of the third side is half the measure of each equal side, then find the measure of the third side.
Ans: Given that the perimeter of an isosceles triangle is \(15 \mathrm{~cm}\)
Let the measure of each equal side of an isosceles triangle be \(x \mathrm{~cm}\).
Then, the measure of the third side will be \(\frac{x}{2} \mathrm{~cm}\).
We know that the perimeter of any polygon is the total length of the boundary of a polygon.
Then, \(x+x+\frac{x}{2}=15 \mathrm{~cm}\)
\(\Rightarrow \frac{5 x}{2}=15 \mathrm{~cm}\)
\(\Rightarrow x=6 \mathrm{~cm}\)
Third side \(=\frac{x}{2}=3 \mathrm{~cm}\)
Hence, the measure of the third side of an isosceles triangle is \(3 \mathrm{~cm}\).

Q.3. One of the acute angles of a right-angle triangle is \(50^{\circ}\). Find the other acute angle.
Ans: Given that the measure of one of the acute angles of a right-angle triangle is \(50^{\circ}\).
Take the measure of the other acute angle as \(x\).
We know that the measure of one of the angles in a right-angle triangle is \(90^{\circ}\).

Now, \(50^{\circ}+90^{\circ}+x=180^{\circ}\) (the sum of the measure of interior angles of a triangle is \(\left.180^{\circ}\right)\)
\(\Rightarrow 50^{\circ}+90^{\circ}+x=180^{\circ}\)
\(\Rightarrow 140^{\circ}+x=180^{\circ}\)
\(\Rightarrow x=180^{\circ}-140^{\circ}=40^{\circ}\)
Hence, the measure of the other acute angle is \(40^{\circ}\).

Q.4. If the perimeter of a scalene triangle is \(12 \,\text {cm}\). The lengths of the two sides are \(3 \,\text {cm}\) and \(4 \,\text {cm}\). Find the measure of the third side.
Ans: Let the measure of the third side be \(x\).
So, \(x+3+4=12 \mathrm{~cm}\)
\(\Rightarrow x=(12-3-4) \mathrm{cm}\)
\(\Rightarrow x=5 \mathrm{~cm}\)
Hence, the measure of the third side is \(5 \mathrm{~cm}\).

Q.5. If the measure of the vertex angle of an isosceles triangle is \(120^{\circ}\), then find the measure of equal angles.
Ans: Given that the measure of the vertex angle of an isosceles triangle is \(120^{\circ}\)
Let the measure of each equal angle be \(x\).

Now, \(120^{\circ}+x+x=180^{\circ}\)
\(\Longrightarrow 2 x=180^{\circ}-120^{\circ}=60^{\circ}\)
\(\Longrightarrow x=30^{\circ}\)
Hence, the measure of each equal angle is \(30^{\circ}\).

Summary

In this article, we learnt about the definition of a triangle, types of triangles, types of triangles based on sides, types of a triangle based on angles, solved examples on the classification of triangles and FAQs on the classification of triangles.

The learning outcome of this article is that you will learn about the fundamental properties of many sorts of triangles and how to determine other elements when just one or two elements of a triangle are given.

FAQs on Classification of Triangles

Q.1. What are the 7 types of a triangle?
Ans: The seven types of triangle are equilateral triangle, scalene triangle, isosceles triangle, acute-angled triangle, right-angled triangle, right isosceles triangle and obtuse-angled triangle.

Q.2. What is the classification of a triangle?
Ans: Classification of triangles are based on the measure of their angles and sides:
a) Based on side lengths: equilateral triangle, scalene triangle, and isosceles triangle.
b) Based on angle measures: acute-angled triangle, right-angled triangle, right-isosceles triangle, obtuse-angled triangle.

Q.3. How do you classify triangles by side lengths?
Ans: Triangles are grouped into three groups based on the length of their sides:
1. Scalene triangle
2. Isosceles triangle
3. Equilateral triangle

Q.4. What is a scalene triangle?
Ans: A scalene triangle is a triangle with varied lengths on all three sides. A scalene triangle’s angles are also measured differently.

Q.5. What are the 2 basic properties of a triangle?
Ans: The \(2\) basic properties of a triangle:
1. Angle sum property of a triangle: According to the angle sum property, the sum of a triangle’s three internal angles is always \(180^\circ \).
2. Exterior angle property: The exterior angle of a triangle is always equal to the sum of the two interior opposing angles.

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