Ex.7.2 Q3 Coordinate Geometry Solution - NCERT Maths Class 10

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Question

To conduct Sports Day activities, in your rectangular shaped school ground \(ABCD\), lines have been drawn with chalk powder at a distance of \(1\rm\,m \) each. \(100\) flower pots have been placed at a distance of \(1\rm\,m \) from each other along \(AD\), as shown in the following figure. Niharika runs \(\begin{align}\frac{1}{4}\end{align}\)th the distance \(AD\) on the \(2^\rm {nd}\) line and posts a green flag. Preet runs \(\begin{align}\frac{1}{5}\end{align}\)th the distance \(AD\) on the eighth line and posts a red flag. What is the distance between both the flags? If Rashmi has to post a blue flag exactly halfway between the line segments joining the two flags, where should she post her flag?

 

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 Known?

  • The school ground \(ABCD\) is rectangular shaped.
  • Lines are drawn at a distance of \(1\rm\,m \) each and \(100\) flower pots have been placed at a distance of \(1\rm\,m \) each along \(AD\).
  • The distance covered by Niharika and Preet on line \(AD\).

What is Unknown?

  • The distance between the flags posted by Niharika and Preet.
  • The position on the line segment where Rashmi has to post the flag.

Steps:

From the Figure,

Given,

  • By observation, that Niharika posted the green flag at of the distance \(P\) i.e., \(\begin{align}\left( {\frac{1}{4} \times 100} \right){\text{m}} = 25\,{\text{m}}\end{align}\) from the starting point of \(2^\rm{nd}\) Therefore, the coordinates of this point \(P\) is \((2, 25)\).
  • Similarly, Preet posted red flag at \(\frac{{1}}{5}\) of the distance \(Q\) i.e., \(\begin{align}\left( {\frac{1}{5} \times 100} \right){\text{m}} = 20\,{\text{m}}\end{align}\) from the starting point of \(8^\rm {th}\) Therefore, the coordinates of this point \(Q\) are \((8, 20)\)

We know that the distance between the two points is given by the Distance Formula,

\(\begin{align}\sqrt {{{\left( {{{\text{x}}_1} - {{\text{x}}_2}} \right)}^2} + {{\left( {{{\text{y}}_1} - {{\text{y}}_2}} \right)}^2}} & & ...\,{\text{Equation}}\,(1)\end{align}\)

To find the distance between these flags \(PQ\) by substituting the values in Equation (1),

\(\begin{align}{\text{PQ}} &= \sqrt {{{(8 - 2)}^2} + {{(25 - 20)}^2}} \\ &= \sqrt {36 + 25} \\ &= \sqrt {61{\text{m}}} \end{align}\)

  • The point at which Rashmi should post her blue flag is the mid-point of the line joining these points.
  • Let this point be \(M \;(x, y)\).

By Section formula

\(\begin{align}  {\text{P(x,}}\,{\text{y)}}& = \left[ {\frac{{{\text{m}}{{\text{x}}_2} + {\text{n}}{{\text{x}}_1}}}{{{\text{m}} + {\text{n}}}},\;\frac{{{\text{m}}{{\text{y}}_2} + {\text{n}}{{\text{y}}_1}}}{{{\text{m}} + {\text{n}}}}} \right] & & ...\;  \end{align}\)  Equation (2) 

\(\begin{align}{\text{x}} &= \frac{{2 + 8}}{2}, \qquad {\text{y}} = \frac{{25 + 20}}{2}\\{\text{x}} &= \frac{{10}}{2}, \qquad\quad\; {\text{y}} = \frac{{45}}{2}\\{\text{x}} &= 5, \qquad\qquad {\text{y}} = 22.5\end{align}\)

Therefore, Rashmi should post her blue flag at \(22.5\;\rm m\) on \(5^\rm {th}\) line