Ex.7.2 Q6 Coordinate Geometry Solution - NCERT Maths Class 10

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Question

If \((1, 2)\), \((4, y)\), \((x, 6)\) and \((3, 5)\) are the vertices of a parallelogram taken in order, find \(x\) and \(y\).

 

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.

\(\begin{align}{\text{P}}({\text{x}},{\text{y}}) = \left[ {\frac{{{\text{m}}{{\text{x}}_2} + {\text{n}}{{\text{x}}_1}}}{{{\text{m}} + {\text{n}}}},\frac{{{\text{m}}{{\text{y}}_2} + {\text{n}}{{\text{y}}_1}}}{{{\text{m}} + {\text{n}}}}} \right] & & \end{align}\)

What is the known?

The \(x\) and \(y\) co-ordinates of the vertices of the parallelogram.

What is the unknown?

The missing \(x\) and \(y\) co-ordinate.

Steps:

From the Figure,

Given,

  • Let \(A \;(1, 2)\), \(B \;(4, y)\), \(C\;(x, 6)\), and \(D \;(3, 5)\) are the vertices of a parallelogram \(ABCD\).
  • Since the diagonals of a parallelogram bisect each other, Intersection point \(O\) of diagonal \(AC\) and \(BD\) also divides these diagonals

Therefore, \(O\) is the mid-point of \(AC\) and \(BD\).

If \(O\) is the mid-point of \(AC\), then the coordinates of \(O\) are

\(\begin{align}\left( {\frac{{1 + {\text{x}}}}{2},\;\frac{{2 + 6}}{2}} \right) \Rightarrow \left( {\frac{{{\text{x}} + 1}}{2},\;4} \right)\end{align}\)

If \(O\) is the mid-point of \(BD\), then the coordinates of \(O\) are

\(\begin{align}\left( {\frac{{4 + 3}}{2},\;\frac{{5 + {\text{y}}}}{2}} \right) \Rightarrow \left( {\frac{7}{2},\;\frac{{5 + {\text{y}}}}{2}} \right)\end{align}\)

Since both the coordinates are of the same point \(O\),

\(\begin{align} &\therefore \frac{{{\text{x}} + 1}}{2} = \frac{7}{2}{\text{ and }}\;4 = \frac{{5 + {\text{y}}}}{2}  \end{align}\)

\(\begin{align} &\Rightarrow {\text{x}} + 1 = 7\;\;{\text{and}}\;\;5 + {\text{y}} = 8 \,   \end{align}\) (By cross multiplying & transposing)

\(\begin{align}  &\Rightarrow {\text{x}} = 6{\text{ and y}} = 3\end{align}\)