If A (-14, -10), B(6, -2) is given, find the coordinates of the points which divide segment AB into four equal parts
If A (-14, -10), B(6, -2) is given, find the coordinates of the points which divide segment AB into four equal parts
Let P \[\left( x_1 , y_1 \right)\] and Q \[\left( x_2 , y_2 \right)\] divide the line AB into 3 equal parts.
AP = PQ = QB
\[\frac{AP}{PB} = \frac{AP}{PQ + QB} = \frac{AP}{AP + AP} = \frac{AP}{2AP} = \frac{1}{2}\]
So, P divides AB in the ratio 1 : 2.
\[x_1 = \frac{1 \times \left( - 4 \right) + 2 \times 2}{2 + 1} = 0\]
\[ y_1 = \frac{1 \times \left( - 8 \right) + 2 \times 7}{2 + 1} = 2\]
\[\left( x_1 , y_1 \right) = \left( 0, 2 \right)\]
Also, \[\frac{AQ}{QB} = \frac{AP + PB}{QB} = \frac{QB + QB}{QB} = \frac{2}{1}\]
\[x_2 = \frac{2 \times \left( - 4 \right) + 1 \times 2}{2 + 1} = - 2\]
\[ y_2 = \frac{2 \times \left( - 8 \right) + 1 \times 7}{2 + 1} = - 3\]
\[\left( x_2 , y_2 \right) = \left( - 2, - 3 \right)\]
Thus, the coordinates of the point of intersection are
P \[\left( x_1 , y_1 \right) = \left( 0, 2 \right)\]
Q \[\left( x_2 , y_2 \right)\] =\[\left( - 2, - 3 \right)\]
Chapter 5. Co-ordinate Geometry – Practice Set 5.2 (Page 115)