Question

In a circle whose radius is $8$cm, a chord is drawn at a point $3$ cm. from the centre of the circle. The chord is divided into two segments by a point on it. If one segment of the chord is $9$ cm, What is the length of the other segment?

Answer 1

Let $O$ be the center of the circle and $OA=OB$ are radius and $AB$ is the chord. We know perpendicular from center, bisects the chord.

$M$ is midpoint of $AB$ such that $\angle OMA=\angle OMB$

and $AM=BM$

In $△AOM,$ $(AM{)}^2=(OA{)}^2-\left(OM{)}^2\right(AM{)}^2=8^2-3^2=64-9AM=\sqrt{55}$

$AB=2AM=2\sqrt{55}cm$

Let a point $P$ on $AB$ such that $AP=9cm$

$BP=AB-AP=2\sqrt{55}-9=5.832cm$