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Math Help - Circle Geometry Help Please

  1. #1
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    Question Circle Geometry Help Please

    A (-6,2) and B (-4,-2) are endpoints of a chord of a circle. C (2.-4) and D (8,4) are endpoints of a second chord.

    a) Determine the coordinates of the center of the circle.

    b) Determine the radius of the circle.

    So lost, please help
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  2. #2
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    Hi

    The center of the circle is equidistant from A and B therefore is on the mediator of [AB]
    It is also on the mediator of [CD]
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  3. #3
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    Quote Originally Posted by PiratePrincess View Post
    A (-6,2) and B (-4,-2) are endpoints of a chord of a circle. C (2.-4) and D (8,4) are endpoints of a second chord.

    a) Determine the coordinates of the center of the circle.

    b) Determine the radius of the circle.

    So lost, please help
    An alternative way is...

    The centre of the circle is the same distance away from all 4 points.

    If the centre is (x,y) then we can use the distance formula
    on any 3 of the 4 points to find (x,y).

    [x-(-6)]^2+[y-2]^2=[x-(-4)]^2+[y-(-2)]^2=[x-2]^2+[y-(-4)]^2

    (x+6)^2+(y-2)^2=(x+4)^2+(y+2)^2=(x-2)^2+(y+4)^2

    Multiply this out

    x^2+12x+36+y^2-4y+4=x^2+8x+16+y^2+4y+4=x^2-4x+4+y^2+8y+16

    Since x^2+y^2 is common to all three, we can eliminate them

    12x-4y+40=8x+4y+20=-4x+8y+20

    Combining the first two equations to write x in terms of y

    12x-8x+40=4y+4y+20\ \Rightarrow\ 4x+40=8y+20\ \Rightarrow\ x+10=2y+5\ \Rightarrow\ x=2y-5

    Using this in the 2nd and 3rd equations

    8(2y-5)+4y+20=-4(2y-5)+8y+20

    16y-40+4y+20=-8y+20+8y+20

    20y-20=40

    20y=60\ \Rightarrow\ y=3\ \Rightarrow\ x=6-5=1

    The radius is the distance from the centre to any of the 4 points

    r=\sqrt{(8-1)^2+(4-3)^2}=\sqrt{7^2+1^2}=\sqrt{50}
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  4. #4
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    Smile Thank you!

    Thank you so, so much! I wish I could hug you!
    I'm a Mom of three, taking a grade 12 math class at night. I haven't been in a classroom in over 15 years and I'm finding it very challenging (to put it nicely lol).
    This forum has been invaluable to me.
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  5. #5
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    circle geometry help please

    Quote Originally Posted by PiratePrincess View Post
    A (-6,2) and B (-4,-2) are endpoints of a chord of a circle. C (2.-4) and D (8,4) are endpoints of a second chord.

    a) Determine the coordinates of the center of the circle.

    b) Determine the radius of the circle.

    So lost, please help

    running back several days I noticed this problem and took a crack at it but
    unfortunately discovered that your two lines are chords of different circles
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  6. #6
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    Quote Originally Posted by bjhopper View Post
    running back several days I noticed this problem and took a crack at it but
    unfortunately discovered that your two lines are chords of different circles
    Distance from (1,3) to (-6,2) is \sqrt{7^2+1^2}

    Distance from (1,3) to (-4,-2) is \sqrt{5^2+5^2}

    Distance from (1,3) to (2,-4) is \sqrt{1^2+7^2}

    Distance from (1,3) to (8,4) is \sqrt{7^2+1^1}

    In all cases the distance is \sqrt{50}

    hence both are chords of the same circle centred at (1,3) with radius \sqrt{50}
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  7. #7
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    circle geometry

    [quote=Archie Meade;469199]Distance from (1,3) to (-6,2) is \sqrt{7^2+1^2}

    Distance from (1,3) to (-4,-2) is \sqrt{5^2+5^2}

    Distance from (1,3) to (2,-4) is \sqrt{1^2+7^2}

    Distance from (1,3) to (8,4) is \sqrt{7^2+1^1}

    In all cases the distance is \sqrt{50}

    hence both are chords of the same circle centred at (1,3) with radius \sqrt{50}[/quote

    Sorry I'm wrong. Iplotted the points and got slightly different radii lenghts and although th center was 1,3 i should have calculated all the radii
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