Question

Twelve synchronized swimmers are forming a circle. The location of those swimmers are (13,-2) (-1,-2) and (6,-9). A 4th swimmer will appear in the center of the circle. Where would the center swimmers need to be located?

Answer 1

The swimmer will need to be at the point (6, -2).

The point (6, -2) is equidistant from and in the middle of all three of the points. Therefore, it would be the center of the circle. You can graph them and plot a circle through the points to be sure. 

Answer 2

The location of the 4th swimmer which is appear in the center of the circle made by twelve synchronized swimmers is (6,-2)

What is the equation of circle?

The equation of the circle is the equation which is used to represent the circle in the algebraic equation form with the value of centre point in the coordinate plane and measure of radius.

The standard form of the equation of the circle can be given as,

$x^2+y^2+2gx+2fy+c=0$

Here, (-g,-f) are coordinate of the center of the circle.

The location of the first swimmer is (13,-2). Put the values in above equation as,

$13^2+(-2)^2+2g(13)+2f(-2)+c=0\\169+4+26g-4f+c=0\\26g-4f+c+173=0$     ........1

Similarly, the equation when we put the value of the location of second swimmer (-1-2) become,

$(-1)^2+(-2)^2+2g(-1)+2f(-2)+c=0\\1+4-2g-4f+c=0\\-2g-4f+c+5=0$........2

The location of the third swimmer is (6,-9). Put the values in above equation as,

$6^2+(-9)^2+2g(6)+2f(-9)+c=0\\36+81+12g-18f+c=0\\12g-18f+c+117=0$        ........3

On solving all the three equation, we get,

$g=-6\\f=2\\c=-9$

Thus, the center point will locate at,

$(-g,-f)=(-(-6),-2)\\(-g,-f)=(6,-2)$

Hence, the location of the 4th swimmer which is appear in the center of the circle made by twelve synchronized swimmers is (6,-2)

Learn more about the equation of circle here;

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