Ex.7.2 Q9 Coordinate Geometry Solution - NCERT Maths Class 10

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Question

Find the coordinates of the points which divide the line segment joining \(A \;(-2, 2)\) and \(B\;(2, 8)\) into four equal parts.

Text Solution

Reasoning:

The coordinates of the point \(P(x, y)\) which divides the line segment joining the points \(A(x1, y1)\) and \(B(x2, y2)\), internally, in the ratio \(\rm m1 : m2\) is given by the Section Formula.

What is the known?

The \(x\) and \(y\) co-ordinates of the points \(A\) and \(B\).

What is the unknown?

The coordinates of the points which divide the line segment joining \(A \;(-2, 2)\) and \(B\;(2, 8)\) into four equal parts.

Steps:

From the Figure,

By observation, that points \(P\), \(Q\), \(R\) divides the line segment \(A\; (-2, 2)\) and \(B \;(2, 8)\) into four equal parts

Point \(P\) divides the line segment \(AQ\) into two equal parts

Hence, Coordinates of

\[\begin{align}  P &= \left( {\frac{{1 \times 2 + 3 \times ( - 2)}}{{1 + 3}},\;\frac{{1 \times 8 + 3 \times 2}}{{1 + 3}}} \right)  \\&= \left( { - 1,\;\frac{7}{2}} \right)\end{align}\]

Point \(Q\) divides the line segment \(AB\) into two equal parts

Coordinates of

\[\begin{align} Q &= \left( {\frac{{2 + ( - 2)}}{2},\;\frac{{2 + 8}}{2}} \right)\\ &= (0,\;5)\end{align}\]

Point \(R\) divides the line segment \(BQ\) into two equal parts

Coordinates of

\[\begin{align} R &= \,\,\left( {\frac{{3 \times 2 + 1 \times ( - 2)}}{{3 + 1}},\;\frac{{3 \times 8 + 1 \times 2}}{{3 + 1}}} \right)\;\\ &=\left(1, \frac{13}{2}\right)\end{align}\]