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Question 5
Students of a school are standing in rows and columns in their playground for a drill practice. A,B,C and D are the positions of four students as shown in figure. 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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Solution

Yes, from the figure we observe that the positions of four students A, B, C and D are (3,5), (7,9), (11,5) and (7,1) respectively i.e. these are four vertices of a quadrilateral. Now, we will find the type of this quadrilateral. For this, we will find length of all of its sides.

Now, AB=(73)2+(95)2[by distance formula,d=(x2x1)2+(y2y1)2]AB=(4)2+(4)2=16+16AB=42BC=(117)2+(59)2=(4)2+(4)2=16+16=42CD=(711)2+(15)2=(4)2+(4)2=16+16=42And DA=(37)2+(51)2=(4)2+(4)2=16+16=42
Now, we can see that AB = BC = CD = DA. i.e., all sides are equal.
Now, we will find length of both the diagonals,
AC=(113)2+(55)2=(8)2+0=8and BD=(77)2+(19)2=0+(8)2=8
Here,AC = BD
Since,AB = BC = CD = DA and AC = BD
The playground represents a square.
Now, the diagonals of a square bisect each other.
So, P (midpoint of diagonal) will be the position of Jaspal so that he is equidistant from each of the four students A, B, C and D.
Coordinates of point P = Mid-point of AC.
(3+112,5+52)(142,102)(7,5)⎢ ⎢ ⎢ Midpoint of line segment having points (x1,y1) and (x2,y2)=(x1+x22,y1+y22)⎥ ⎥ ⎥
Hence, the required position of Jaspal is (7,5)



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