Question

9. Find the points of trisection of the line segment joining the points A(2,-2) and
B(-7, 4).

Answer 1

Hey!
We know that, The points of trisection divides the line in three equal parts.

Also, If a point P(x,y) divides a line joined by (a, b) , ( c, d) into the ratio m:n then ,
x = mc + na / m + n
y = md + nb / m + n

Given two points, A(2,-2) & B ( -7,4)

Let P, Q are the points of trisection of the line AB.( Here P is closer to A)

We know that, AP = PQ = QB.
So, P divides the line segment AB internally in the ratio 1 : 2 .

Let co ordinates of P = ( g, h)

So, g = 1(-7) + 2(2) / 3 = -7 + 4/3 = -1 .

h = 1(4) + 2(-2) / 1 +2 = 4 - 4 /3 = 0/3 = 0 .

Now, Q divides the line segment AB internally in the ratio ( 2 : 1 )

Co ordinates of Q ( u, v)

Now,
u = 2 ( -7) + 1 (2) / 3 = -12/3 = -4

v = 2( 4 ) + 1( -2) / 3 = 8 -2 / 3 = 2 .

Therefore, The points of trisection are ( -1 , 0 ) , ( -4 , 2 )

Hope helped!

Answer 2

Answer,

Let P and Q are the points of the intersection of the line segment joining the points A and B.
Here, AP = PQ = QB.

AP = 1PQ = 1QB = 1

By section formula,

(lx2 + mx1 / l + m , ly2 + my1 / l + m)

P divides the line segment AB In ratio , 1:2
where, l = 1 , m = 2

A = (2 , -2 )
B = (-7 , 4)

{ ( 1 + (-7) + ( 2 × 2 ) / 1 + 2 , (1 × 4) + ( 2 × - 2) /1+2}

(-7 + 4 / 3 ) , ( 4 - 4 /3 )

(-3/3 , 0/3 )

(-1/0)

Q divides line segments AB In the ratio , 2:1
where, l = 2 , m = 1

A = (2,-2)
B = ( -7,4)

{ (2 × (-7) + ( 1 × 2 ) / 2 + 1 , ( 2 × 4 ) + ( 1 × -2 )/2 + 1}

( - 14 + 2 / 3 , 8 - 2 / 3 )

- 12 /3 , 6 / 3

-4/2

Therefore,

The coordinates are, P ( -1 , 0) and Q ( -4 ,2 ) .