The Geometry of a Circle - By: Dennis Kapatos

9) What is the equation of the circle for the sprinkler at (0,0), before it’s setting was
changed?
100 2 = x 2 + y 2
10) Name 4 points that lay on the edge of this circle.
(100,0), (0,100), (-100,0), (0,-100)
Bob decides that changing the range of all the sprinklers will create too much overlap and
thus waste too much water (water is expensive in this part of the country), so he decides
to install a new sprinkler at the point (100,58).
(0,0)
(100,0)
11) If this new sprinkler can exactly reach point (100,43) then what is its spraying radius?
Use the distance formula and show your work.
r = √( (100-100) 2 + (58-43) 2 ) = 15 meters
12) If this new sprinkler can exactly reach point any point (x,y) then what is its spraying
radius? (Note: This will be an equation in terms of r, x, and y.)
√( (100 - x) 2 + (58- y) 2 )
r = __________________________
r 2 = (100 - x) 2 + (58- y) 2
Now rewrite this without a square root sign: ______________________________
13) Inside the pair of parentheses of this equation should be minus signs. If the x or y
come after the minus sign then switch them with the other number. Write this equation.
__________________________________
r 2 = (x - 100) 2 + (y - 58) 2
14) What is this the same as doing and why is it allowed in this case? That is, why
doesn’t it change the validity of the equation in this case? Convince yourself that this
doesn’t change anything before moving on.
Reversing the order is the same as multiplying everything
inside the parentheses by a -1, which, because the result is
squared afterwards, doesn’t effect the equation.
(a - b) 2 = (-1 (a - b)) 2 = (-a + b) 2 = (b – a) 2