Question

Two concentric circles have a common centre O a line L intersectthe outer circle at A and B and the inner circle at C and D . If AB =x and CD = y prove that AC = BD = $\frac{1}{2}$ (x - y)

Answer 1

Let O be the common centre of both the circles. AB and CD are chords of outer and inner circles respectively.
Draw OM perpendicular to line ACDB.
OM bisects both the chords AB and CD.
$\to AM=BM=\frac{x}{2}andCM=DM=\frac{y}{2}AC=AM-CM=\frac{x}{2}-\frac{y}{2}=\frac{1}{2}(x-y)$
Similarly $BD=BM-DM=\frac{x}{2}-\frac{y}{2}=\frac{1}{2}(x-y)$
So, $AC=BD=\frac{1}{2}(x-y)$ Proved.