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Points $P, Q, R$ and $S$ divide the line segment joining the points $A (1, 2)$ and $B (6, 7)$ in 5 equal parts. Find the coordinates of the points $P, Q$ and $R$.
Given:
Points $P, Q, R$ and $S$ divide the line segment joining the points $A (1, 2)$ and $B (6, 7)$ in 5 equal parts.
To do:
We have to find the coordinates of the points $P, Q$ and $R$.
Solution:
Let the coordinates of $P, Q, R, S$ be $(x_1,y_1), (x_2,y_2), (x_3,y_3), (x_4,y_4)$ respectively.
Points $P, Q, R$ and $S$ divide the line segment joining the points $A (1, 2)$ and $B (6, 7)$ in 5 equal parts.
This implies,
$AP = PQ = QR = RS = SB$
Using the section formula, if a point $( x,\ y)$ divides the line joining the points $( x_1,\ y_1)$ and $( x_2,\ y_2)$ in the ratio $m:n$, then
$(x,\ y)=( \frac{mx_2+nx_1}{m+n},\ \frac{my_2+ny_1}{m+n})$
This implies,
\( \mathrm{P} \) divides \( \mathrm{AB} \) in the ratio \( 1: 4 \)
\( (x_{1}, y_{1})=(\frac{1 \times 6+4 \times 1}{1+4}, \frac{1 \times 7+4 \times 2}{1+4}) \)
\( =( \frac{6+4}{5}, \frac{7+8}{5}) \)
\( =(\frac{10}{5}, \frac{15}{5}) \)
\( =(2, 3) \)
The coordinates of \( \mathrm{P} \) are \( (2,3) \).
\( \mathrm{Q} \) divides \( A B \) in the ratio \( 2: 3 \).
\( (x_{2}, y_{2})=(\frac{2 \times 6+3 \times 1}{2+3}, \frac{2 \times 7+3 \times 2}{2+3}) \)
\( =(\frac{12+3}{5}, \frac{14+6}{5}) \)
\( =( \frac{15}{5}, \frac{20}{5}) \)
\( =(3, 4) \)
The coordinates of \( \mathrm{Q} \) are \( (3,4) \).
\( \mathrm{R} \) divides \( \mathrm{AB} \) in the ratio \( 3: 2 \).
\( (x_{3}, y_{3})=(\frac{3 \times 6+2 \times 1}{3+2}, \frac{3 \times 7+2 \times 2}{3+2}) \)
\( =(\frac{18+2}{5}, \frac{21+4}{5}) \)
\( =(\frac{20}{5}, \frac{25}{5}) \)
\( =(4, 5) \)
The coordinates of \( \mathrm{R} \) are \( (4,5) \).
\( \mathrm{S} \) divides \( \mathrm{AB} \) in the ratio \( 4: 1 \).
\( (x_{4}, y_{4})=(\frac{4 \times 6+1 \times 1}{4+1}, \frac{4 \times 7+1 \times 2}{4+1}) \)
\( =(\frac{24+1}{5}, \frac{28+2}{5}) \)
\( =(\frac{25}{5}, \frac{30}{5}) \)
\( =(5, 6) \)
The coordinates of \( \mathrm{S} \) are \( (5,6) \).
The coordinates of the points $P, Q$ and $R$ are $(2,3), (3,4)$ and $(4,5)$ respectively.
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