Altitude And Median Of A Triangle

Triangle - A triangle is a three-sided closed polygon which has three sides and three vertices. It consists of three angles. When three line segments are joined together they form a two-dimensional structure known as triangle.

Types Of Triangle

1. Based on the measurement of the angles of the triangle

• Acute Angled Triangle

In this type of triangle all the angles of the triangle are less than 90°

• Obtuse Angled Triangle

In this type of triangle any one angle of the triangle is more than 180°

• Right Angled Triangle

In this type of triangle one angle of the triangle is equal to 90°

2. On the Basis of the length of the sides of the triangle

• Scalene Triangle

In this type of triangle, the length of all the three sides is different due to which the measurement of all the angles is different.

• Isosceles Triangle

In this type of triangle, the length of any two sides of the triangle is equal due to which the angles corresponding to the two equal sides are equal.

• Equilateral Triangle

In this type of triangle, the length of all three sides is equal and due to this all the angles in the triangle are equal and are equal to 60°.

Properties Of A Triangle

• Three vertices, three angles, and three sides make up a triangle.

• Any two triangle sides when added together have a length larger than the third side.

• Difference between the lengths of the two sides of the triangle is less than the third side.

• Sum of all the angles of a triangle is 180°, also known as the angle sum property.

• Sum of all the exterior angles of a triangle is 360°.

• An exterior angle of a triangle is equal to the sum of the internal opposite angles, known as exterior angle property of triangle.

• The side opposite to the largest angle of the triangle is the largest side of that angle.

Pythagoras Theorem

It states that “the square of the hypotenuse is equal to the sum of the square of the other two sides”. Hypotenuse^2=Base^2+Altitude^2

Perimeter Of Triangle

The total length of all three sides of a triangle is known as the perimeter of the triangle or the sum of all three sides of the triangle is known as the perimeter of the triangle.

Perimeter = Sum of all the three sides

Area Of Triangle

The region occupied by a triangle in the 2D space is known as area of the triangle.

Area of Triangle = \frac{1}{2} \times base \times height

Heron’s Formula

Triangle area is calculated using Heron's formula by

A=\sqrt{s(s-a)(s-b)(s-c)}

Where s is the semi-parameter of the triangle

s=\frac{(a+b+c)}{2}

Congruent Triangle

Two triangles are said to be congruent if the corresponding sides and the corresponding angles are equal of both the triangles.

Types of Congruency:-

• Angle Angle Side – AAS

• Angle Side Angle – ASA

• Side Side Side – SSS

• Side Angle Side – SAS

• Right Angled Hypotenuse Side – RHS

Altitude Of Triangle

Line drawn from the vertex of a triangle and ends at the opposite side making a right angle is known as the altitude of the triangle.

Properties Of Altitude Of Triangle

• A Triangle has a maximum of three altitudes, one from each vertex.

• It is the shortest distance between the vertex angle and the opposite side of the triangle.

• All three altitudes meet at a single point inside the triangle always, this point is known as the orthocentre of the triangle.

• It is possible that the altitude of the triangle may lie inside and outside of the triangle.

• It is used to calculate the area of the triangle together with the base to which the altitude is drawn.

Median Of Triangle

A line segment drawn from the vertex and joining the midpoint of the opposite side of the triangle is known as the median of the triangle.

Properties Of Median Of Triangle

• A Triangle has a maximum of three medians, one from each vertex.

• The triangle is split into two triangles with equal areas by the median.

• All three medians meet at a single point inside the triangle always; this point is known as the centroid of the triangle.

• All three medians of the triangle divide the complete triangle into six smaller triangles of equal area.

Different Points Inside The Triangle:-

1. Incentre

It is the intersection point of all the three angle bisectors drawn from all the three interior angles of the triangle.

2. Circumcentre

It is the intersection point of all three perpendicular bisectors drawn from the three sides of the triangle.

3. Centroid

It is the intersection point of all three medians of the triangle.

4. Orthocentre

It is the intersection point of all three altitudes of the triangle.