# geometry

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Two chords intersect a circle. The shorter chord is divided into segments of lengths of 9 inches and 12 inches. The longer chord has a length of 24 inches. Find the length of the shorter portion of the longer chord .

• geometry -

make a sketch
angles subtended by the same chord are equal,
so you have two similar triangles
let the 24 inch chord be divided into x and 24-x
then by ratios ...
x/12 = 9/24-x
108 = 24x - x^2
x^2 - 24x + 108 = 0
(x-6)(x-18)=0
x = 6 or 18

if one piece is 6, the other is 24-6 = 18

the shorter piece is 6

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