Properties of A Triangle

A triangle is a polygon with three vertices. A vertex is a point where two or more lines or edges converge. In the case of a triangle, the three vertices are joined by three line segments called "sides" or "legs". A triangle is usually referred to by its vertices. 

Types of triangles 

Triangles can be classified according to the lengths of their sides. The three most common types of triangles are the equilateral triangle, the isosceles triangle, and the scalene triangle. An equilateral triangle has all sides of equal lengths. An isosceles triangle has two sides of equal lengths. A scalene triangle has a different length on each side.

Equality and Congruence 

For any given angle, the degree measure of that angle determines its relationship to the other angles in the same triangle. The sum of the angles of a triangle is 180 degrees. If two angles in a triangle have the same measure, then the two triangles are said to be congruent. Triangles can also be classified as acute, right, or obtuse, depending on the measure of their greatest angle.

In an equilateral triangle, all sides are of equal length and the angles measure 60 degrees each. An isosceles triangle has two sides of equal length and two angles of equal measure. The sides of a scalene triangle have different lengths and the angles of a scalene triangle have different measures.

Relative to its respective sides, an angle in a triangle is said to be acute if it measures less than 90 degrees. An angle in a triangle is obtuse if it measures greater than 90 degrees and less than 180 degrees. 

Introduction

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 it is 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, point I which is the meeting point of the bisectors of 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. A circumcircle is a circle drawn keeping the circumcentre of the triangle as the centre such that the circle passes through all the vertices of the triangle.

Properties of a Triangle

Angle sum property of Triangle: Angle Sum Property of a Triangle states that the sum of all the three angles of a triangle is equal to 180 degrees.

Exterior angle Property: Each side of a triangle can be extended both ways. The three sides of a triangle give rise to six extended sides, each side making to extended sides. An exterior angle is an angle between one side and one extended side of a triangle. There are three extended sides in a triangle and each extended side results in one exterior angle, therefore, a triangle has six exterior angles. In the diagram given below, angles 1, 2, 3, 4, 5 and 6 are exterior angles of the triangle. 

Note: The angles between two extended sides are not exterior angles. 

Exterior Angle Property states that:

1. The sum of any two interior angles of a triangle is equal to the opposite exterior angle.

2. The sum of the equivalent exterior angle of a triangle is always equal to 360 degrees.                                        

Properties of Similar Triangles

Similar triangles have the same basic shapes. Triangles having the same angles but different sizes are called Similar Triangles as the ratio of the sides remains the same. The rotated triangles or the mirror image triangles are also called Similar triangles as the angles and sizes are the same.

• The ratio of the corresponding sides of two similar triangles is the same.

• The corresponding angles of two similar triangles are congruent.

Similar Triangles can be identified in three ways:

• Angle Angle Angle: Two triangles are said to be similar triangles if all the three sides of one triangle are equal to the corresponding angles of the other triangle.

• Side Side Side: Two Triangles are said to be similar if the sides of one triangle are equal or in the same ratio to the corresponding sides of another triangle.

• Side Angle Side: Two triangles are said to be similar if the two sides of one triangle are equal or in a ratio to the corresponding sides of the other triangle provided the included angles are also congruent.

Median Property of a Triangle

• It bisects the angle at the vertex of an isosceles and equilateral triangle whose adjacent sides are equal.

• It also bisects the angle of the vertex of an isosceles and equilateral triangle.

• There are only three medians of a triangle.

• The point of intersection of the medians is called the centroid.

• It divides the area of a triangle into two halves.

• The length of each median is divided in the ratio 2:1 by the centroid.

• The centroid divides the triangle into 6 smaller triangles of equal area. 

• All the medians of equilateral triangles are equal.

• The medians drawn from vertices of an isosceles triangle with equal angles are equal in length.

• The length of all the medians of a scalene triangle is different.

• Apollonius Theorem can be used to find the length of the medians.

Properties of Circumcenter of a Triangle

• The centre of the circumcircle is the circumcentre of the triangle.

• The vertices are at an equal distance from the circumcenter of a triangle.

• For the acute-angled triangle, the circumcenter is always inside the triangle.

• For the obtuse-angled triangle, the circumcentre is outside the triangle.

• For the right-angled triangle, the Circumcenter is at the midpoint of the hypotenuse.

Properties of Circumcentre of a triangle

• It is the point of intersection of the perpendicular bisectors of the sides of a triangle 

• It is equidistant from the vertices of the triangle.                                          

Properties of Angle Bisector of a triangle

• A point on the angle bisector of the triangle is equidistant from its sides. 

• There is always a common point at which the angle bisectors of a triangle meet.

• It proves the congruency between two angles.                         

Properties of Incentre of a triangle

• It is the intersection point of the angle bisector of a triangle

• It is equidistant from the sides of the triangle.                                  

Properties of the centroid of a triangle

• It is the intersection of three medians of a triangle.

• It is a point of congruency of a triangle.

• It is always on the inside of a triangle.

• It acts as the centre of gravity of a triangle.

• It divides each median in the ratio 2:1.

Classification of Triangles on the basis of sides and their properties

Scalene Triangle: A triangle in which all the sides are of different lengths is called Scalene Triangle. In the adjoining figure, all the sides of the triangle are of different lengths (i.e, 13, 9, and 14 cm).

Properties

• A scalene triangle has all the sides different.

• It has unequal angles.

• Its longest side is right opposite to its biggest angle.

• It cannot be bisected into two equal halves. 

• It has no line of symmetry.

• A scalene triangle can be an acute scalene triangle, an obtuse scalene triangle or a right scalene triangle.

• The area of a scalene triangle can be calculated by using Heron’s formula if all the sides are given.

• When a scalene triangle is inscribed in a circle, each angle is half the angle subtended by the opposite side. 

• The centre of the circumscribing circle lies inside the triangle if all three angles are acute.

Isosceles Triangle: A triangle having two sides of equal length is called the Isosceles triangle. In an isosceles triangle, the angles opposite to the equal sides are equal. In the triangle given below, two sides are of 5inches and one side is 3 inches. Thus, it is an Isosceles Triangle.   

Properties:

• Two sides are congruent to each other.

• The unequal side of an isosceles triangle is called a base.

• The two angles opposite to the equal sides are congruent to each other. Thus it has two congruent base angles.

• Apex angle is the angle that is not congruent to the two congruent base angles.

• The height drawn from the apex of an isosceles triangle divides the base into two equal parts and also divides the apex angle into two equal angles.

• Area of Isosceles triangle = ½ × base × height

• The perimeter of an Isosceles triangle = sum of all the three sides

• The third unequal angle of an isosceles can be acute or obtuse.

• The circumcenter of an isosceles triangle lies inside the triangle if all the three angles of the three triangles are acute. 

• The sides of the triangle are the chords of the circumcircle.

• If one of the angles is 90 degrees, then the circumcenter lies outside the triangle. 

• The centroid is the intersection of the medians of the Isosceles triangle. 

• The median drawn from Apex divides the triangle at right angles.

• The perpendicular bisectors of an isosceles triangle intersect at its circumcenter.

• The angle bisectors of an isosceles triangle intersect at the incenter.

• The circle drawn with the incenter touches the three sides of the triangle internally.

• Each median divides the isosceles triangle into two equal triangles having the same area.

• The area of the triangle can be estimated:

• If the measure of one angle and one side are given

•  If three sides of the triangle are given.

• If two sides of an isosceles triangle and their included angles are given.

• Joining the midpoint of three sides divides the triangle into 4 smaller triangles of the same area.

• When a circle with a diameter equal to the base is drawn:

• For an obtuse-angled isosceles triangle, the apex lies inside the circle. 

• For a right-angled isosceles triangle, the apex lies on the circumference. 

• For an acute-angled isosceles triangle, the apex lies outside the triangle.

• When the midpoint apex is taken as a radius and a circle is drawn with the midpoint of the base as the centre 

• For acute-angled isosceles, the base vertices lie inside the circle.

• For a right-angled isosceles the base vertices lie on the circumference

• For an obtuse-angled isosceles triangle, the base vertices lie outside the circle.

Equilateral Triangle

A triangle whose all three sides are of equal length is called an equilateral triangle. The measure of each angle of a triangle is 60 degrees. The triangle given below is equilateral because all the sides are equal (i.e, 4cm) with angles 60 degrees each.

Properties:

• All the sides of an equilateral triangle are always the same.

• All the angles of the equilateral triangle are the same.

• The median, altitude, angle bisector, perpendicular bisector all coincide at one line.

• The median, altitude, perpendicular bisector and angle bisector forms the line of symmetry of an equilateral triangle.

• The length of all the medians, altitude, perpendicular bisector, and angle bisector is the same.

• The area of an equilateral triangle is 23 /4 s2. Here, s is the sides of an equilateral triangle.

• The orthocenter, circumcenter, incenter, and centroid all lie at the same point. 

• Each altitude is a median of the equilateral triangle.

• The centroid is the meeting point of the angle bisectors, medians as well as perpendicular bisectors of a triangle.

• The incenter and the circumcenter of an equilateral triangle are the same.

• The area of an equilateral triangle can be estimated:

• If the measure of one angle and one side are given

• If three sides of the triangle are given.

• If two sides of the triangle and their included angles are given.

Classification of Triangles based on angles and their properties

Acute Triangle: A triangle in which every angle measures more than zero degrees but less than 90 degrees is called an acute-angled Triangle. All the interior angles of the triangle given below are 30, 70 and 80 degrees i.e, all are less than 90 degrees.      

Properties:

• It has all three angles as acute.

• Its perpendicular bisectors intersect at the circumcenter and the median intersect at the centroid.

• The circumcenter of an acute triangle lies inside the triangle.

• The angle bisectors meet at the incenter of the circle. A circle can be drawn with the incenter of the triangle as the centre of the circle to touch the three sides of the triangle internally.

• Joining the midpoints of the three sides of the triangle results in 3 parallelograms having the same area and 4 triangles of the same area.

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. In the given triangle, one angle is 120 degrees which are greater than 90 degrees whereas the other two angles are less than 90 degrees.     

Properties:

• It has two acute angles and one obtuse angle.

• Its perpendicular bisectors intersect at the circumcenter.

• The medians intersect at the centroid.

• Its circumcenter always lies outside the triangle.

• The angle bisectors meet at the incenter of the triangle. A circle is drawn with the incentre as its centre touches the three sides of the triangle internally.

• Medians divide the triangle into two smaller triangles having the same area.

• Joining the midpoints of the three sides of the triangle results in 3 parallelograms of the same area and 4 triangles of the same area.

Right Triangle: A triangle in which one of the measures of the angles exactly 90 degrees is called a right triangle. In the triangle given below, one of the three angles of the triangle is 90 degrees whereas the other two angles are less than 90 degrees.

Properties:

• Out of the three angles of a right-angled triangle, one angle is greater than 90 degrees and the other two are acute angles.

• The longest side of the right triangle is called the hypotenuse and the angle opposite to the hypotenuse is 90 degrees.

• The area of a right triangle is half the product of the base and height.

• The radius of the circumcircle is always half the hypotenuse and the centre of the circumcircle is always the midpoint of the hypotenuse.

• A-line perpendicular to the hypotenuse from the right angle results in three similar triangles.