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 segment 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 = `(x_3 + 8)/2`

∴ 12 = `(x_3 + 8)/2`

∴ 24 = (x3 + 8)

∴ x3 = 16

y co-ordinate of Q = `(y_3 + 16)/2`

∴ 14 = `(y_3 + 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,

x co-ordinate of P = `(x_1 + 12)/2`

∴ 16 = `(x_1 + 12)/2`

∴ 32 = x1 + 12

∴  x1 = 20

y co-ordinate of P = `(y_1 + 14)/2`

∴ 12 = `(y_1 + 14)/2`

∴ 24 = y1 + 14

∴ y1 = 10

∴ A(x1, y1) = (20, 10)

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

By midpoint formula,

x co-ordinate of S = `(x_2 + 8)/2`

∴ 4 = `(x_2 + 8)/2`

∴ 8 = x2 + 8

∴  x2 = 0

y co-ordinate of S = `(y_2 + 16)/2`

∴ 18 = `(y_2 + 16)/2`

∴ 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.