Question

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?

Answer 1

`r=sqrt85` unit

Explanation:

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