Question

The line segment AB is divided into five congruent parts at P, Q, R and S such that A-P-Q-R-S-B. If point Q (12, 14) and S (4, 18) are given, find the coordinates of A, P, R, B

Answer 1

Points P, Q, R and S divide seg AB in five congruent parts. 

Let A (x₁ , y₁ ), B (x₂ , y₂ ), P (x₃ , y₃ ) and R (x₄ , y₄ ) be the given points. 

Point R is the midpoint of seg QS. 

By midpoint formula,

x co-ordinate of R = (12 + 4)/2 = 16/2 = 8 

y co-ordinate of R = (14 + 18)/2  = 32/2 = 16 

∴ co-ordinates of R are (8, 16). 

Point Q is the midpoint of seg PR. 

By midpoint formula,

x co-ordinate of Q = (x₃ + 8)/2

∴ 12 = (x₃ + 8)/2

∴ 24 = x₃ + 8

∴  x₃ = 16

y co-ordinate of Q = (y₃ + 16)/2

∴ 14 = (y₃ + 16)/2

∴ 28 = y + 16 

∴ y₃ = 12 

∴ P(x₃ ,y₃ ) = (16, 12) 

∴ co-ordinates of P are (16, 12). Point P is the midpoint of seg AQ. 

By midpoint formula,

∴ co-ordinates of A are (20, 10). Point S is the midpoint of seg RB. 

By midpoint formula,

∴ 36 = y₂ + 16 

∴ y₂ = 20 

∴ B(x₂ , y₂ ) = (0, 20) 

∴ co-ordinates of B are (0, 20). 

∴ The co-ordinates of points A, P, R and B are (20, 10), (16, 12), (8, 16) and (0, 20) respectively.