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 \(A(x_1, y_1)\) and \(B(x_2, y_2)\), internally, in the ratio \(m_1 : m_2\) 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

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