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There are four points that are 5 units from the line y=13 and 13 units from the point (7,13). What is the sum of the x and y and coordinates of all four of these points?

Nov 28, 2020

#1
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Not as difficult as it seems once we  draw  a graph :  https://www.desmos.com/calculator/wars2lfhv7

Construct a circle with a center of (7,13)  and  a  radius of 13

And consider the line y  = 13

This line will pass through (7,13)

The equation of our circle is

(x-7)^2  + ( y - 13)^2  =  169

Let y = 18   ( this will be 5 vertical  units  from y = 13 )

(x - 7)^2  + (18 - 13)^2  =  169

(x - 7)^2  + 5^2 =  169

(x - 7)^2  + 25 =  169

(x - 7)^2  =  169 -25

(x -7)^2  = 144

12^2 = 144          and          (-12)^2  =  144

So  x must be  19       or      x =  -5

And one of  the points we  need is  ( 19, 18)This point will lie on our circle and will be 13 units from (7 ,13)  and it will be 5 units  from the line  y = 13

And one ot the other points we  need is  (-5, 18)

And using symmetry.....the other two points we  need are     (19, 13 - 5)  = (19, 8)     and (-5,13-5) = (-5,8)   Nov 28, 2020
#2
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Thank you so much!!

MoonWolfie  Nov 28, 2020
#3
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OK, MoonWolfie....I forgot to sum the  coordinates, but I suppose you  can handle that....      CPhill  Nov 28, 2020
edited by CPhill  Nov 28, 2020