Question
Find the coordinates of the points which divide the line segment joining A(-2, 2) and B(2, 8) into four equal parts.
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Answer
Let $P (x_1, y_1) Q(x_2, y_2)$ and $R(x_3, y_3)$ be the points which divide the line segment AB into four equal parts.
Then, P divides AB in the ratio 1 : 3 internally.
$x=\frac{mx_2+nx_1}{m+n}$
$\therefore x _ { 1 } = \frac { ( 1 ) ( 2 ) + ( 3 ) ( - 2 ) } { 1 + 3 }$
$= \frac { 2 - 6 } { 4 } = - \frac { 4 } { 4 } = - 1$
$y=\frac{my_2+ny_1}{m+n}$
$y _ { 1 } = \frac { ( 1 ) ( 8 ) + ( 3 ) ( 2 ) } { 1 + 3 }$
$= \frac { 8 + 6 } { 4 } = \frac { 14 } { 4 } = \frac { 7 } { 2 }$
So, $\mathrm { P } \rightarrow \left( - 1 , \frac { 7 } { 2 } \right)$
Also, Q divides AB in the ratio 1 : 1 i.e.
Q is the mid point of AB
$x _ { 2 } = \frac { - 2 + 2 } { 2 } = 0$
$y _ { 2 } = \frac { 2 + 8 } { 2 } = \frac { 10 } { 2 } = 5$
So, $Q \rightarrow ( 0,5 )$
and, R divides AB in the ratio 3 : 1
$\therefore x _ { 2 } = \frac { ( 3 ) ( 2 ) + ( 1 ) ( - 2 ) } { 3 + 1 }$
$= \frac { 6 - 2 } { 4 } = \frac { 4 } { 4 } = 1$
$y _ { 3 } = \frac { ( 3 ) ( 8 ) + ( 1 ) ( 2 ) } { 3 + 1 }$
$= \frac { 24 + 2 } { 4 } = \frac { 26 } { 4 } = \frac { 13 } { 2 }$
So, $\mathrm { R } \rightarrow \left( 1 , \frac { 13 } { 2 } \right)$
Then, P divides AB in the ratio 1 : 3 internally.
$x=\frac{mx_2+nx_1}{m+n}$
$\therefore x _ { 1 } = \frac { ( 1 ) ( 2 ) + ( 3 ) ( - 2 ) } { 1 + 3 }$
$= \frac { 2 - 6 } { 4 } = - \frac { 4 } { 4 } = - 1$
$y=\frac{my_2+ny_1}{m+n}$
$y _ { 1 } = \frac { ( 1 ) ( 8 ) + ( 3 ) ( 2 ) } { 1 + 3 }$
$= \frac { 8 + 6 } { 4 } = \frac { 14 } { 4 } = \frac { 7 } { 2 }$
So, $\mathrm { P } \rightarrow \left( - 1 , \frac { 7 } { 2 } \right)$
Also, Q divides AB in the ratio 1 : 1 i.e.
Q is the mid point of AB
$x _ { 2 } = \frac { - 2 + 2 } { 2 } = 0$
$y _ { 2 } = \frac { 2 + 8 } { 2 } = \frac { 10 } { 2 } = 5$
So, $Q \rightarrow ( 0,5 )$
and, R divides AB in the ratio 3 : 1
$\therefore x _ { 2 } = \frac { ( 3 ) ( 2 ) + ( 1 ) ( - 2 ) } { 3 + 1 }$
$= \frac { 6 - 2 } { 4 } = \frac { 4 } { 4 } = 1$
$y _ { 3 } = \frac { ( 3 ) ( 8 ) + ( 1 ) ( 2 ) } { 3 + 1 }$
$= \frac { 24 + 2 } { 4 } = \frac { 26 } { 4 } = \frac { 13 } { 2 }$
So, $\mathrm { R } \rightarrow \left( 1 , \frac { 13 } { 2 } \right)$
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60-80
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