SOLUTION: Find the radius of a circle with center at (4,1) if a chord of length 4 times square root of 2 is bisected at (7,4).

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Question 166147: Find the radius of a circle with center at (4,1) if a chord of length 4 times square root of 2 is bisected at (7,4).
Answer by Edwin McCravy(6938) About Me  (Show Source):
You can put this solution on YOUR website!
Find the radius of a circle with center at (4,1) if a chord of length 4 times square root of 2 is bisected at (7,4).

Let O be the center of the circle (4,1).
Let P be the point (7,4), 
Let A and B be the endpoints of the chord,
so the chord is the segment AB. Draw the
graph:

drawing%28400%2C400%2C-3%2C12%2C-8%2C7%2C%0D%0A%0D%0Alocate%285%2C6.5%2CA%29%2C+locate%289.2%2C2.5%2CB%29%2C+locate%284%2C1%2CO%29%2C+%0D%0Alocate%287%2C4.6%2CP%29%2C%0D%0A%0D%0Agraph%28400%2C400%2C-3%2C12%2C-8%2C7%29%2C%0D%0A%0D%0Acircle%284%2C1%2Csqrt%2826%29%29%2C+line+%285%2C6%2C9%2C2%29%2C+%0D%0A%0D%0Aline%283.9%2C1%2C4.1%2C1%29%2C+line%284%2C.9%2C4%2C1.1%29%2C%0D%0A%0D%0Aline%286.9%2C4%2C7.1%2C4%29%2C+line%287%2C3.9%2C7%2C4.1%29%0D%0A%0D%0A%29

Draw in OA and OP:

drawing%28400%2C400%2C-3%2C12%2C-8%2C7%2C%0D%0A%0D%0Alocate%285%2C6.5%2CA%29%2C+locate%289.2%2C2.5%2CB%29%2C+locate%284%2C1%2CO%29%2C+%0D%0Alocate%287%2C4.6%2CP%29%2C%0D%0Atriangle%284%2C1%2C7%2C4%2C5%2C6%29%2C%0D%0Agraph%28400%2C400%2C-3%2C12%2C-8%2C7%29%2C%0D%0A%0D%0Acircle%284%2C1%2Csqrt%2826%29%29%2C+line+%285%2C6%2C9%2C2%29%2C+%0D%0A%0D%0Aline%283.9%2C1%2C4.1%2C1%29%2C+line%284%2C.9%2C4%2C1.1%29%2C%0D%0A%0D%0Aline%286.9%2C4%2C7.1%2C4%29%2C+line%287%2C3.9%2C7%2C4.1%29%0D%0A%0D%0A%29

We need to find the length of OA since it 
is a radius of the circle.

Triangle OPA is a right triangle, since if a
bisector of a chord passes through the center of
a circle, then it is perpendicular to the chord.

We are given that chord AB has length 4sqrt%282%29,
and since P bisects it, then AP is half that length
or 2sqrt%282%29.

Next we will find the length of OP by use of the
distance formula:

d=sqrt%28+%28x%5B2%5D-x%5B1%5D%29%5E2+%2B+%28y%5B2%5D-y%5B1%5D%29%5E2+%29

using the given coordinates of O(4,1) and P(7,4):

OP=sqrt%28+%287-4%29%5E2+%2B+%284-1%29%5E2+%29

OP=sqrt%28+%283%29%5E2+%2B+%283%29%5E2+%29

OP=sqrt%289+%2B+9+%29

OP=sqrt%2818%29

OP=3sqrt%282%29

Now by the Pythagorean theorem,

OA%5E2=AP%5E2%2BOP%5E2

OA%5E2=%282sqrt%282%29%29%5E2%2B%283sqrt%282%29%29%5E2

OA%5E2=%284%2A2%29%2B%289%2A2%29

OA%5E2=8%2B18

OA%5E2=26

OA=sqrt%2826%29

And since OA is a radius, the circle has 
radius sqrt%2826%29.

Edwin