# What is the Median and Altitude of a Triangle

## What is the Median and Altitude of a Triangle

A closed figure bounded by three line segments is called a triangle.

It is a 3-sided polygon and is named as ‘ΔABC’.

In the above figure:

(a) Number of sides forming ΔABC are 3, i.e., AB, BC, and CA.

(b) Number of vertices (i.e., initial and terminal points of sides) are 3, i.e., A, B, and C.

(c) Number of angles are 3, i.e., ∠A, ∠B, and ∠C.

(d) Sum of the angles of a triangle is 180°, i.e., ∠A + ∠B + ∠C = 180°.

The side AB is called the base line of ΔABC and the angle formed at vertex C opposite the base line AB is called the vertical angle.

### Interior and exterior of a triangle

A triangle drawn on a plane divides the plane into 3 parts interior, exterior, and the triangle itself.

The points lying inside ΔABC (i.e., cross) forms the interior of ΔABC and the points lying outside ΔABC (i.e., dots) forms the exterior of the ΔABC. Linear boundary of the triangle is the triangle itself.

### Altitude of a triangle

An altitude of a triangle is the line segment drawn from a vertex of a triangle, perpendicular to the line containing the opposite side.

(i) PS is an altitude on side QR in figure.

(ii) AD is an altitude, with D the foot of perpendicular lying on BC in figure.

(iii) The side PQ, itself is an altitude to base QR of right angled PQR in figure.

Note:

(i) A triangle has three altitudes.

(ii) All the three altitudes meet at a point H (called orthocentre of triangle) i.e., all altitudes of any triangle are concurrent.

(iii) Orthocentre of the triangle may lie inside the triangle [Figure (i)],, outside the triangle [Figure (ii)] and on the triangle [Figure (iii)].

Orthocentre

The point of concurrence of the altitudes of a triangle is called the orthocentre of the triangle.

Notes :

1. Since the altitudes of a triangle are concurrent, therefore to locate the orthocentre of a triangle, it is sufficient to draw its two altitudes.
2. Although altitude of a triangle is a line segment, but in the statement of their concurrence property, the term altitude means a line containing the altitude (line segment).
 Properties of Altitudes Properties of Orthocentre 1. The altitudes of an equilateral triangle are equal. 1. The orthocentre of an acute-angled triangle lies in the interior of the triangle. 2. The altitude bisects the base of an equilateral triangle. 2. The orthocentre of a right-angled triangle is the vertex containing the right angle. 3. The altitudes drawn on equal sides of an isosceles triangle are equal. 3. The orthocentre of an obtuse-angled triangle lies in the exterior of the triangle.

### Median of a triangle

A line segment that joins a vertex of a triangle to the mid-point of the opposite side is called a median of the triangle.

For example, consider DLMN. Let S be the mid-point of MN, then LS is the line segment joining vertex L to the mid point of its opposite side.

The line segment LS is said to be the median of DLMN.

Similarly, RN and MT are also medians of DLMN.

Note :

(i)   A triangle has three medians.

(ii)  All the three medians meet at  one point G (called centroid of the triangle) i.e., all medians of any triangle are concurrent.

(iii) The centroid of the triangle always lies inside of triangle.

(iv) The centroid of a triangle divides each one of the medians in the ratio 2 : 1.

(v)  The medians of an equilateral triangle are equal in length.

Example: The angles of a triangle ABC are in the ratio of 1 : 2 : 3. Find all the angles of ΔABC.

Solution: Let the first angle A be x.

∴ ∠B = 2x and ∠C = 3x

Sum of angles of a triangle = 180°

x + 2x + 3x = 180°

6x = 180°

x = 30°

So, ∠A = 30°

∠B = 2 × 30° = 60°,

∠C = 3 × 30° = 90°.