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Find the coordinates of the points of trisection of the line segment joining (4, -1) and (-2, -3).

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#### Solution

Let P (x_{1}, y_{1}) and Q (x_{2}, y_{2}) are the points of trisection of the line segment joining the given points i.e., AP = PQ = QB

Therefore, point P divides AB internally in the ratio 1:2.

∴ `x = (m_1x_2 + m_2x_1)/(m_1 + m_2),` `y = (m_1y_2 + m_2y_1)/(m_1 + m_2)`

`x_1= (1xx(-2)+2xx4)/(1+2), y_1 = (1xx(-3)+2xx(-1))/(1+2)`

`x_1 = (-2+8)/3=6/3=2, y_1 = (-3-2)/3 = (-5)/3`

Therefore P(x_{1},y_{1}) = `(2, -5/3)`

Point Q divides AB internally in the ratio 2:1.

∴ `x = (m_1x_2 + m_2x_1)/(m_1 + m_2),` `y = (m_1y_2 + m_2y_1)/(m_1 + m_2)`

`x_2=(2xx(-2)+1xx4)/(2+1), y_2=(2xx(-3)+1xx(-1))/(2+1)`

`x_2 = (-4+4)/3 = 0, y_2= (-6-1)/3 = (-7)/3`

`Q(x_2, y_2) = (0, -7/3)`

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