Question

i) Find the coordinates of the point which divides the line segment join- ing the points (4, -3) and (8,5) in the ratio 3:1 internally. ii) Find the mid point of the line segment joining the points (3, 0) and (-1, 4).​

Answer 1

Step-by-step explanation:

Solutions:-

I)

Given points are (4, -3) and (8,5)

Let A = (4,-3)

=>(x1, y1) = (4,-3)=>x1 = 4 and y1 = -3

Let B = (8,5)

=> (x2, y2) =(8,5) => x2 = 8 and y2 = 5

Given ratio = 3:1

=> m1:m2 =3:1 => m1 = 3 and m2 = 1

Let the coordinates of the point which divides the linesegment joining the given points be P(x,y) .

We know that

The coordinates of the point P(x,y) which divides the linesegment joining the points (x1,y1) and (x2,y2) in the ratio m1:m2 internally is ({m1x2+m2x1}/(m1+m2), {m1y2+m2y1}/(m1+m2))

On Substituting these values in the above formula then

=>({(3)(8)+(1)(4)}/(3+1) , {(3)(5)+(1)(-3)}/(3+1))

=> ({24+4)}/4 , {15-3}/4)

=> ( 28/4 , 12/4 )

=> (7,3)

The coordinates of the required point is (7,3)

ii)

Given points are (3,0) and (-1,4)

Let A = (3,0)

=>(x1, y1) = (3,0)=>x1 = 3 and y1 = 0

Let B = (-1,4)

=> (x2, y2) =(-1,4) => x2 = -1 and y2 = 4

The coordinates of the mid point of the linesegment joining the points A (x1, y1) and

B (x2, y2) is M(x,y) =((x1+x2)/2 , (y1+y2)/2)

On Substituting these values in the above formula then

=> M(x,y) = ( ( 3+(+1))/2 , (0+4)/2)

=> M(x,y) = ((3-1)/2, 4/2)

=> M(x,y) = (2/2,4/2)

=> M(x,y) = (1,2)

Answer:-

I) The coordinates of the required point is (7,3)

ii) The coordinates of the required mid point is (1,2)

Used formulae:-

Section Formula :-

The coordinates of the point P(x,y) which divides the linesegment joining the points (x1,y1) and (x2,y2) in the ratio m1:m2 internally is ( {m1x2+m2x1}/(m1+m2), {m1y2+m2y1}/(m1+m2) )

Mid point Formula :-

linesegment joining the points (x1, y1) and

(x2, y2) is M(x,y) = ( (x1+x2)/2 , (y1+y2)/2)