Chords In Concentric Circles

Geometry Level 3

Let $O_1$ and $O_2$ be concentric circles with radii $4$ and $6$ , respectively. A chord $AB$ is drawn in $O_1$ with length $2$ . Extend $AB$ to intersect $O_2$ in points $C$ and $D$ . If the length of $CD$ can be expressed as $a\sqrt{b}$ , where $a$ and $b$ are positeve integers, find the value of $a+b$ .

The answer is 23.

1 solution

Alan Guo
Nov 27, 2015

Let the midpoint of AB and CD be X, and O be the centre of the circles. Then $OX\perp AB, CD$ , and $AX=XB=1$ .

By Pythagoras', $OX^2 = OA^2-AX^2$ $CX^2 = OC^2 - OX^2$ $= 6^2 - (4^2 - 1^2)$ $=21$ Thereofore $CD = 2CX = 2\sqrt{21}$ , and so $a+b = 2 + 21 = 23$ .

Chords In Concentric Circles

The answer is 23.

1 solution

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