Circle chord

Determine the radius of the circle in which the chord 6 cm away from the center of the circle is 12 cm longer than the radius of the circle.

Correct result:

r =  14.58 cm

Solution:

 r2=62+(r+122)2  3r224r288=0  a=3;b=24;c=288 D=b24ac=24243(288)=4032 D>0  r1,2=b±D2a=24±40326=24±2476 r1,2=4±10.5830052443 r1=14.5830052443 r2=6.58300524426   Factored form of the equation:  3(r14.5830052443)(r+6.58300524426)=0  r>0  r=14.58 cm \ \\ r^2 = 6^2+(\dfrac{ r + 12 }{2})^2 \ \\ \ \\ 3r^2 -24r -288 =0 \ \\ \ \\ a=3; b=-24; c=-288 \ \\ D = b^2 - 4ac = 24^2 - 4\cdot 3 \cdot (-288) = 4032 \ \\ D>0 \ \\ \ \\ r_{1,2} = \dfrac{ -b \pm \sqrt{ D } }{ 2a } = \dfrac{ 24 \pm \sqrt{ 4032 } }{ 6 } = \dfrac{ 24 \pm 24 \sqrt{ 7 } }{ 6 } \ \\ r_{1,2} = 4 \pm 10.5830052443 \ \\ r_{1} = 14.5830052443 \ \\ r_{2} = -6.58300524426 \ \\ \ \\ \text{ Factored form of the equation: } \ \\ 3 (r -14.5830052443) (r +6.58300524426) = 0 \ \\ \ \\ r>0 \ \\ \ \\ r = 14.58 \ \text{cm}



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