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

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Last updated date: 16th Jun 2024
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Answer
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Hint: To solve this question, we will use the concept of section formula. The coordinates of the point R which divides the line segment joining two points \[P\left( {{x_1},{y_1}} \right)\] and \[Q\left( {{x_2},{y_2}} \right)\] in the ratio m:n are given by, $x = \dfrac{{m{x_2} + n{x_1}}}{{m + n}}$ and \[y = \dfrac{{m{y_2} + n{y_1}}}{{m + n}}\]

Complete step-by-step answer:
Given that,
A line segment joining the points (1, -2) and (-3, 4) gets trisected and we have to find the coordinates of those points which trisects this line.
We know that the line segment which gets trisects means the line is divided either into 2:1 or in 1:2.
So,
Let A and B be the points which trisects the line PQ. Then, AP = AB = BQ.
Therefore, A divides the line PQ in the ratio 1:2 and B divides the line PQ in the ratio 2:1.
Case 1: when $A\left( {x,y} \right)$ divides the line in ratio 1:2.
By using the section formula,
The coordinates of the point $A\left( {x,y} \right)$ is given by,
\[ \Rightarrow A\left( {x,y} \right) = \left( {\dfrac{{1 \times \left( { - 3} \right) + 2 \times 1}}{{1 + 2}},\dfrac{{1 \times \left( 4 \right) + 2 \times \left( { - 2} \right)}}{{1 + 2}}} \right)\]
\[ \Rightarrow A\left( {x,y} \right) = \left( {\dfrac{{ - 3 + 2}}{3},\dfrac{{4 - 4}}{3}} \right)\]
\[ \Rightarrow A\left( {x,y} \right) = \left( {\dfrac{{ - 1}}{3},0} \right)\]
Case 2: when $B\left( {x,y} \right)$ divides the line in 2:1.
By using the section formula,
The coordinates of the point $B\left( {x,y} \right)$ is given by,
\[ \Rightarrow B\left( {x,y} \right) = \left( {\dfrac{{2 \times \left( { - 3} \right) + 1 \times 1}}{{2 + 1}},\dfrac{{2 \times \left( 4 \right) + 1 \times \left( { - 2} \right)}}{{2 + 1}}} \right)\]
\[ \Rightarrow B\left( {x,y} \right) = \left( {\dfrac{{ - 6 + 1}}{3},\dfrac{{8 - 2}}{3}} \right)\]
\[ \Rightarrow B\left( {x,y} \right) = \left( {\dfrac{{ - 5}}{3},2} \right)\]
Hence, we can say that the coordinates of the points which trisects the line segment joining (1, -2) and (-3, 4) are \[\left( {\dfrac{{ - 1}}{3},0} \right)\] are \[\left( {\dfrac{{ - 5}}{3},2} \right)\]

Note: In this type of questions, we also have to remember that the coordinates of the mid-point of the line segment joining by the two points \[P\left( {{x_1},{y_1}} \right)\] and \[Q\left( {{x_2},{y_2}} \right)\] are given by, $\left( {\dfrac{{{x_1} + {x_2}}}{2},\dfrac{{{y_1} + {y_2}}}{2}} \right)$
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