(i) (5, -6) and (- 7, 5),
Let our given points be A(5,-6) and B(-7, 5) and required points be C (x1 , y1 ) and D(x2 , y2)
The points of trisection of a line are points which divide into the ratio 1:2
By section formula,
x = \(\frac{mx_2 + nx_1}{m + n},\) y = \(\frac{my_2 + ny_1}{m + n}\)
For point C(x1 , y1 )
x1 = \(\frac{1\times(-7)+2\times5}{1 + 2}, \)
y1 = \(\frac{1\times5+2\times(-6)}{1 + 2}, \) …Here m = 1 and n = 2
∴ x1 = \(\frac{3}3\), y1 = \(\frac{-7}3\)
∴ C (x1 , y1 ) ≡ \((1,\frac{-7}3)\)
For points D (x2 , y2)
x2 = \(\frac{1\times(-7)+1\times5}{2 + 1}, \)
y2 = \(\frac{2\times5+1\times(-6)}{2 + 1}\)…Here m = 2 and n = 1
∴ x2 = \(\frac{-9}3\), y2 = \(\frac{4}3\)
∴ D (x2 , y2) ≡ \((- 3,\frac{4}3)\)
Hence, the points of trisection of line joining given points are
\((1,\frac{-7}3)\) and \((-3,\frac{4}3)\)
(ii) (3, -2) and (-3, -4)
Let our given points be A(3,-2) and B(-3, -4) and required points be C (x1 , y1 ) and D(x2 , y2)
The points of trisection of a line are points which divide into the ratio 1:2
By section formula,
∴ x2 = \(\frac{-3}3\), y2 = \(\frac{-10}3\)
∴ D (x2 , y2)≡ \((-1, \frac{-10}3)\)
Hence, the points of trisection of line joining given points are
\((1, \frac{-8}3)\) and \((-1, \frac{-10}3)\)
(iii) (2, -2) and (-7, 4)
Let our given points be A(2,-2) and B(-7, 4) and required points be C (x1 , y1 ) and D(x2 , y2)
The points of trisection of a line are points which divide into the ratio 1:2
By section formula,
∴ x2 = \(\frac{-12}3\), y2 = \(\frac{6}3\)
∴ D (x2 , y2)≡ (-4, 2)
