P divides the segment AB in ratio 1:4

Q divides the segment AB in ratio 2:3

R divides the segment AB in ratio 3:2

For coordinates of P

X = (m₁x₂ + m₂x₁)/ m₁ + m₂

= (1 × 6 + 4 × 1)/1 + 4

= (6 + 4) /5

= 10/5 = 2

Y = (m₁y₂ + m₂y₁)/ m₁ + m₂

= (1x 7 + 4 × 2)/5

= (7 + 8)/5

= 15 / 5 = 3

= (2, 3)

For coordinates of Q

X = (m₁x₂ + m₂x₁)/ m₁ + m₂

= (2x 6 + 3x 1)/5

= (12 + 3) /5

= 15/5 = 3

Y = (m₁y₂ + m₂y₁)/ m₁ + m₂

= (2 × 7 + 3 × 2)/ 5

= (14 + 6)/5

= 20 / 5 = 4

= (3,4)

For coordinates of R

X = (m₁x₂ + m₂x₁)/ m₁ + m₂

= (3 × 6 + 2 × 1)/5

= (18 + 2) /5

= 20/5 = 4

Y = (m₁y₂ + m₂y₁)/ m₁ + m₂

= (3 × 7 + 2 × 2)/ 5

= (21 + 4)/5

= 25 / 5 = 5

= (4,5)

Hence

P(2, 3), Q(3, 4), R(4, 5)