#### 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.

#### Solution

Let the coordinates be

\[B\left( x_4 , y_4 \right) = \frac{8 + x_4}{2} = 4, \frac{16 + y_4}{2} = 18\]

\[ \Rightarrow 8 + x_4 = 8, y_4 = 36 - 16\]

\[ \Rightarrow x_4 = 0, y_4 = 20\]

\[B\left( x_4 , y_4 \right) = \left( 0, 20 \right)\]

\[\frac{AQ}{QB} = \frac{AP + PQ}{QR + RS + SB} = \frac{AP + AP}{AP + AP + AP} = \frac{2}{3}\]

\[\frac{2 \times 0 + 3 \times x_1}{2 + 3} = 12, \frac{2 \times 20 + 3 \times y_1}{2 + 3} = 14\]

\[ \Rightarrow 3 x_1 = 60, 40 + 3 y_1 = 14\]

\[ \Rightarrow x_1 = 20, y_1 = 10\]

\[A\left( x_1 , y_1 \right)\] = (20, 10)

AP = PQ

\[\frac{20 + 12}{2} = x_2 , \frac{10 + 14}{2} = y_2 \]

\[ \Rightarrow \frac{32}{2} = x_2 , \frac{24}{2} = y_2 \]

\[ \Rightarrow x_2 = 16, y_2 = 12\]