if A(1,4) B(2,-3) C(-1,-2) are the vertices of a triangle ABC. find

a) the equation of the median through A

b) the equation of altitude through B

c) the right bisector of BC

Question

if A(1,4) B(2,-3) C(-1,-2) are the vertices of a triangle ABC
find
a) the equation of the median through A
b) the equation of altitude through B
c) the right bisector of BC

Ajanta Trivedi · Accepted Answer

given A(1,4), B(2,-3) and C(-1,-2) are the vertices of the triangle ABC.

(a)

median through A will pass through the mid-point of BC.

let D is the mid-point of BC.

the coordinates of the mid-point of BC is $D (\frac{2 - 1}{2}, \frac{- 3 - 2}{2}) \equiv D (\frac{1}{2}, - \frac{5}{2})$

since the median AD will pass through the points A and D.

the equation of a line passing through two points A(1,4) and $D (\frac{1}{2}, - \frac{5}{2})$ is:

$y - 4 = (\frac{- \frac{5}{2} - 4}{\frac{1}{2} - 1}) \cdot (x - 1) y - 4 = \frac{- \frac{13}{2}}{- \frac{1}{2}} \cdot (x - 1) y - 4 = 13 (x - 1) 13 x - y = - 4 + 13 13 x - y = 9$

therefore the equation of the median through A is 13x-y=9

(b) the equation of the altitude through B is a line perpendicular to AC.

and passes through B.

slope of the straight line AC $m_{1} = \frac{- 2 - 4}{- 1 - 1} = \frac{- 6}{- 2} = 3$

slope of the line perpendicular to AC $m_{2} = - \frac{1}{3}$

since it passes through B(2,-3).

equation of the straight line with slope -1/3 and passes through a given point B(2,-3):

$y + 3 = - \frac{1}{3} (x - 2) 3 y + 9 = - x + 2 x + 3 y + 9 - 2 = 0 x + 3 y + 7 = 0$

therefore the equation of the altitude through B is x+3y+7=0

(c) the right bisector of BC passes through the mid-point of BC and is perpendicular to BC.

the coordinates of the mid-point of BC is $D (\frac{1}{2}, - \frac{5}{2})$ .

the slope of line BC = $= \frac{- 2 + 3}{- 1 - 2} = \frac{1}{- 3} = - \frac{1}{3}$

the slope of the line perpendicular to BC is 3.

equation of the straight line with slope 3 and passes through the given points is:

$y + \frac{5}{2} = 3 (x - \frac{1}{2}) y + \frac{5}{2} = 3 x - \frac{3}{2} 3 x - y = \frac{5}{2} + \frac{3}{2} = \frac{8}{2} 3 x - y = 4$

therefore the equation of the right bisector of BC is 3x-y=4

if A(1,4) B(2,-3) C(-1,-2) are the vertices of a triangle ABC. find a) the equation of the median through A b) the equation of altitude through B c) the right bisector of BC

if A(1,4) B(2,-3) C(-1,-2) are the vertices of a triangle ABC. find

a) the equation of the median through A

b) the equation of altitude through B

c) the right bisector of BC