Question

If A (20, 10), B(0, 20) are given, find the coordinates of the points which divide segment AB into five congruent parts.

Answer 1

Let the points \[P\left( x_1 , y_1 \right), Q\left( x_2 , y_2 \right), R\left( x_3 , y_3 \right) \text { and } S\left( x_4 , y_4 \right)\] be the points which divide the line segment AB into 5 equal parts.

\[\frac{AP}{PB} = \frac{AP}{PQ + QR + RS} = \frac{AP}{4AP} = \frac{1}{4}\]

\[x_1 = \left( \frac{1 \times 0 + 4 \times 20}{1 + 4} \right) = 16\]

\[ y_1 = \left( \frac{1 \times 20 + 4 \times 10}{1 + 4} \right) = 12\]

\[P\left( x_1 , y_1 \right) = \left( 16, 12 \right)\]

\[\frac{PQ}{QB} = \frac{PQ}{QR + RS + SB} = \frac{PQ}{PQ + PQ + PQ} = \frac{PQ}{3PQ} = \frac{1}{3}\]

\[x_2 = \left( \frac{1 \times 0 + 3 \times 16}{1 + 3} \right) = 12\]

\[ y_2 = \left( \frac{1 \times 20 + 3 \times 12}{1 + 3} \right) = 14\]

\[Q\left( x_2 , y_2 \right) = \left( 12, 14 \right)\]

\[\frac{QR}{RB} = \frac{QR}{RS + SB} = \frac{QR}{QR + QR} = \frac{QR}{2QR} = \frac{1}{2}\]

\[x_3 = \left( \frac{1 \times 0 + 2 \times 12}{1 + 2} \right) = 8\]

\[ y_3 = \left( \frac{1 \times 20 + 2 \times 14}{1 + 2} \right) = 16\]

\[R\left( x_3 , y_3 \right) = \left( 8, 16 \right)\]

S is the midpoint of RB so, using the midpoint formula

\[x_4 = \frac{8 + 0}{2} = 4\]

\[ y_4 = \frac{16 + 20}{2} = 18\]

\[S\left( x_4 , y_4 \right) = \left( 4, 18 \right)\]

So, the points 

\[P\left( x_1 , y_1 \right) = \left( 16, 12 \right)\]

\[ Q\left( x_2 , y_2 \right) = \left( 12, 14 \right)\]

\[R\left( x_3 , y_3 \right) = \left( 8, 16 \right)\]

\[S\left( x_4 , y_4 \right) = \left( 4, 18 \right)\]