Students of a school are standing in rows and columns in their playground for a drill practice. $ \mathrm{A}, \mathrm{B}, \mathrm{C} $ and $ \mathrm{D} $ are the positions of four students as shown in figure below. Is it possible to place Jaspal in the drill in such a way that he is equidistant from each of the four students $ A, B, C $ and D? If so, what should be his position?
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Given:

Students of a school are standing in rows and columns in their playground for a drill practice. \( \mathrm{A}, \mathrm{B}, \mathrm{C} \) and \( \mathrm{D} \) are the positions of four students.

To do:

We have to find whether it is possible to place Jaspal in the drill in such a way that he is equidistant from each of the four students \( A, B, C \) and D.

Solution:

From the figure, we observe that the positions of the four students A, B, C and D are $(3,5),(7,9), (11, 5)$ and $(7,1)$ respectively.

The four vertices form a quadrilateral. 

$A B =\sqrt{(7-3)^{2}+(9-5)^{2}}$

$A B =\sqrt{(4)^{2}+(4)^{2}}$

$=\sqrt{16+16}$

$A B =4 \sqrt{2}$

$B C =\sqrt{(11-7)^{2}+(5-9)^{2}}$

$=\sqrt{(4)^{2}+(-4)^{2}}$

$=\sqrt{16+16}$

$=4 \sqrt{2}$

$C D =\sqrt{(7-11)^{2}+(1-5)^{2}}$

$=\sqrt{(-4)^{2}+(-4)^{2}}$

$=\sqrt{16+16}$

$=4 \sqrt{2}$

$D A =\sqrt{(3-7)^{2}+(5-1)^{2}}$

$=\sqrt{(-4)^{2}+(4)^{2}}$

$=\sqrt{16+16}$

$=4 \sqrt{2}$

Here,

$A B=B C=C D=D A$

Therefore,

The position of Jaspal will be the mid point $M(x, y)$ between $AC$

By section formula, 

The mid point $M(x, y)=(\frac{x_{1}+x_{3}}{2}, \frac{y_{1}+y_{3}}{2})$

$M(x, y)=(\frac{3+11}{2}, \frac{5+5}{2})$

$M(x, y)=(7,5)$

Hence, the position(co-ordinates) of Jaspal is $(7,5)$.

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Updated on: 10-Oct-2022

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