Question

Points P, Q and R in that order are dividing a line segment joining A(1, 6) and B (5, -2) in four equal parts. Find the coordinates o P, Q and R.

Answer 1

The given points are A(1,6) and B(5,-2)

Then, P(x,y) is a point that divides the line AB in the ratio 1:3

By section formula

x = (mx2+nx1)/(m+n) , y = (my2+ny1)/(m+n)

⇒x={(1∗5)+(3∗1)}/1+3
⇒x=(5+3)/4
⇒x=8/4
⇒x=2
⇒y={(1∗−2)+(3∗6)}/1+3
⇒y=(−2+18)/4
⇒y=16/4
⇒y=4

Therefore, the co-ordinates of the point P are (2, 4)

Let Q be the midpoint of AB

Then, Q(x,y)

x=(x1+x2)/2

⇒x=(1+5)/2

⇒ x=6/2

⇒ x = 3

y=(y1+y2)/2

⇒ y=(6+(−2))/2

⇒ y=4/2

⇒ y = 2

Therefore the co-ordinates of Q are (3, 2)

Let R (x,y) be the point that divides AB in the ratio 3:1

By section formula

x = (mx2+nx1)/(m+n) , y = (my2+ny1)/(m+n)

⇒x={(3∗5)+(1∗1)}/3+1
⇒x=(15+1)/4
⇒x=16/4
⇒x=4
⇒y={(3∗−2)+(1∗6)}/3+1
⇒y=(−6+6)/4
⇒y=0/4 ⇒y=0

Therefore the co-ordinates of R are (4, 0)

Hence, the coordinates of the points P, Q and R are (2, 4), (3, 2) and (4, 0) respectively.