A chord of a circle divides the circle into two parts such that the squares inscribed in the two parts have areas 16 and 144 square units. the radius of the circle, is?

1 Answer
May 8, 2018

r=sqrt85 unit

Explanation:

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Let O and r be the center and the radius of the circle, respectively, as shown in the figure.
Let AB be the chord,
Side of the small square =sqrt16=4 units
Side of the big square =sqrt144=12 units
Recall that the line joining the center of a circle to the midpoint of a chord is perpendicular to the chord,
=> AB is perpendicular to OC
let OC=x, => OD=x+4
In DeltaODF, OF^2=OD^2+DF^2,
=> r^2=(x+4)^2+2^2
=> r^2=x^2+8x+20 ----- Eq(1)
In DeltaOGH, OH^2=OG^2+GH^2
=> r^2=(12-x)^2+6^2
=> r^2=x^2-24x+180 ----- Eq(2)
Eq(1)=Eq(2)
=> x^2+8x+20=x^2-24x+180,
=> 32x=160, => x=5 units
=> OD=x+4=5+4=9 units,
=> r^2=9^2+2^2=85
=> r=sqrt85 units