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Question

Find the point of tri-section of the line segment joining the points $$(-2,1)$$ and $$(7,4)$$


Solution

Given:- A line segment joining the points $$A \left( -2, 1 \right)$$ and $$B \left( 7, 4 \right)$$.
Let $$P$$ and $$Q$$ be the points on $$AB$$ such that,
$$AP = PQ = QB$$
Therefore, 
$$P$$ and $$Q$$ divides $$AB$$ internally in the ratio $$1 : 2$$ and $$2 : 1$$ respectively.
As we know that if a point $$\left( h, k \right)$$ divides a line joining the point $$\left( {x}_{1}, {y}_{1} \right)$$ and $$\left( {x}_{2}, {y}_{2} \right)$$ in the ration $$m : n$$, then coordinates of the point is given as-
$$\left( h, k \right) = \left( \cfrac{m{x}_{2} + n{x}_{1}}{m+n}, \cfrac{m{y}_{2} + n{y}_{1}}{m+n} \right)$$
Therefore,
Coordinates of $$P = \left( \cfrac{1 \times 7 + 2 \times \left( -2 \right)}{1 + 2}, \cfrac{1 \times 4 + 2 \times 1}{1  + 2} \right) = \left( 1, 2 \right)$$
Coordinates of $$Q = \left( \cfrac{2 \times 7 + 1 \times \left( -2 \right)}{1 + 2}, \cfrac{2 \times 4 + 1 \times 1}{1  + 2} \right) = \left( 4, 3 \right)$$
Therefore, the coordinates of the points of trisection of the line segment joining $$A$$ and $$B$$ are $$\left( 1, 2 \right)$$ and $$\left( 4, 3 \right)$$.

Mathematics

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