Question: A(0,6) and B(10,0) are two points and O is the origin. Find the equation of the median OM and altitude OD of triangle AOB.

Solution:

We need to remember that the altitude of a triangle is perpendicular to its base.

And median is the line that divides the base into two equal parts.

So, median and altitude of triangle is not necessarily the same line in a triangle.

Again, to find the equation of altitude of triangle, we need to remember the following points:
  • Altitude of a triangle is perpendicular to the base.
  • The altitude passes through the opposite vertex of the triangle.
Also, to find the equation of median, we need to remember the following points:
  • Passing point of median is the mid-point of the two vertices of the base.

Given,




∆AOB is a right angled triangle.
O is the origin. Coordinates are O(0,0)
A is a point. Coordinates are A(0,6)
B is a point. Coordinates are B(10,0)
OM is the median.
OD is the altitude of a triangle.

Now,
Let A(0,6) = ($x_1,y_1$) and B(10,0) = ($x_2,y_2$)
Coordinates of midpoint M is
M(x,y) = \left ( \dfrac{x_1+x_2}{2} , \dfrac{y_1+y_2}{2} \right )$
$= \left ( \dfrac{0+10}{2}, \dfrac{6+0}{2} \right )$
$= (5,3)

Let O(0,0) = ($x_1,x_2$) and M(5,3) = ($x_2,y_2$). Equation of line OM is

$or, (y - y_1) = \frac{y_2 - y_1}{x_2 - x_1} (x - x_1)$

$or, (y - 0) = \frac{3-0}{5-0} (x -0)$

$or, y = \dfrac{3x}{5}$

$or, 5y = 3x$

$or, 3x -5y = 0$ is the required equation of OM.


Now,
Let A(0,6) = ($x_1,y_1$) and B(10,0) = ($x_2,y_2$)
Slope of line AB ($m_1$) = $\frac{y_2 - y_1}{x_2 - x_1} $

$= \dfrac{0-6}{10-0}$

$= \dfrac{-6}{10}$

$= \dfrac{-3}{5}$

We know,

OD and AB are perpendicular. So, their slopes are in the following relation.
$m_1 × m_2 = -1$
$or, \frac{-3}{5} × m_2 = -1$
$\therefore m_2 = \frac{5}{3}$

And,

Taking O(0,0) as ($x_1,y_1$) and slope ($m_2$} = \frac{5}{3}, slope of line OD is;

$y - y_1 = m_2 (x - x_1)$

$or, y -0 = \dfrac{5}{3} (x -0)$

$or, 3y = 5x$

$or, 5x -3y = 0$ is the required equation of OD.


Therefore, the required equation of the median OM is 3x -5y = 0 and the required equation of the altitude OD of the triangle is 5x -3y = 0.