Question

Find the co-ordinates of the point of trisection of the line segment joining (4,1) and (-2,-3)​

Answer 1

Step-by-step explanation:

Given :-

The points are (4,1) and (-2,-3)

To find :-

Find the co-ordinates of the point of trisection of the line segment joining (4,1) and (-2,-3) ?

Solution :-

Given points are (4,1) and (-2,-3)

Let A(x1,y1) = (4,1) => x1 = 4 and y1 = 1

Let B(x2, y2) = (-2,-3) => x2 = -2 and

y2 = -3

We know that

The trisectional points divides the given linesegment in the ratio 1:2 or 2:1

Let the Trisectional points of AB linesegment are P and Q

=> AP = PQ = QB

=> AP:PB = 1:2 and AQ:QB = 2:1

(I) On taking the ratio 1:2 :-

Let m1:m2 = 1:2 => m1 = 1 and m2 = 2

We know that

Section formula

The coordinates of the point P which divides the linesegment joining the points (x1, y1) and (x2, y2) in the ratio m1:m2 internally is P(x,y) =

{m1x2+m2x1}/(m1+m2),{m1y2+m2y1}/(m1+m2) )

On Substituting these values in the above formula then

=> P(x,y) = ( {(1×-2)+(2×4)}/(1+2) , {(1×-3)+(2×1)}/(1+2) )

=> P(x,y) = ( (-2+8)/3 , (-3+2)/3 )

= P(x,y) = ( 6/3 , -1/3 )

=> P(x,y) = (2,-1/3)

II) On taking the ratio 2:1:-

Let m1:m2 = 2:1 => m1 = 2 and m2 = 1

We know that

Section formula

The coordinates of the point Qwhich divides the linesegment joining the points (x1, y1) and (x2, y2) in the ratio m1:m2 internally is Q(x,y)

=( {m1x2+m2x1}/(m1+m2),{m1y2+m2y1}/(m1+m2) )

On Substituting these values in the above formula then

=> Q(x,y) = ({(2×-2)+(1×4)}/(2+1), {(2×-3)+(1×1)}/(2+1) )

=> Q(x,y) = ( (-4+4)/3 , (-6+1)/3 )

= Q(x,y) = ( 0/3 , -5/3 )

=> Q(x,y) = (0,-5/3)

Answer:-

The Trisectional points of the linesegment are

(2,-1/3) and (0,-5/3)

Used formulae:-

Section formula:-

→ The coordinates of the point P which divides the linesegment joining the points (x1, y1) and (x2, y2) in the ratio m1:m2 internally is P(x,y)

= ({m1x2+m2x1}/(m1+m2),{m1y2+m2y1}/(m1+m2) )

→ The Trisectional points divides the given linesegment in the ratio 1:2 or 2:1

Trisectional Points :-

The points which divides the given linesegment into three equal parts are called Trisectional points.