Question

Three girls Reshma, Salma and Mandip are playing a game by standing on a circle of radius $5$m drawan in a park. Reshma throws a ball to Salma, Salma to Mandip, Mandip to Reshma. If the distance between Reshma and Salma and between Salma and Mandip is $6m$ each, what is the distance between Reshma and Mandip?

Answer 1

Let Reshma, salma and mandip be donated by point R,S & M respectively

Given radius = $5m$ Ref image

& kJ = $6m,5m=6m$

To find length of RM

construction : Join $OR,OS,OM$

Let RM intersect $OS\cot k$

In $\Delta OKS\&\Delta OMS$

$OR=OM$ (both radius) Ref. image

$OS=OS$ (common)

$RS=SM$ (given)

$\Delta ORS\cong OMS$ (sss conguency)

$\therefore \angle ROS=\angle MOS$ (CPCT) __(1)

$\ln \Delta ORX\&\Delta OMX$

$OR=OM$ (both radius)

$\angle ROX=\angle MOX$ (from (1) $\angle ROS=\angle MOS$)

$OX=OX$ (common)

$\Delta ORX\cong \Delta OMX$ (SAS congsency)

$\therefore RX=MX$ (CPCT __(2))

Since $RX=MX$

so, OX bisects chord RM

$\therefore OX\perp RM$ (line drawn through centre of of circle to bisect a chord as ${\perp }^n$ to the chord)

Ref. image

Now,

Let $OX=\alpha$,

So, $XS=OS-OX=S-\alpha$

From (3) & (4)

Ref. image

$25-x^2=11-x^2+10x$

$25-11-x^2+x^2=10x$

$14=10x$

$10x=14$

$x=1.4$

Putting alue of x in (3)

$R{X}^2=25-x^2$

$R{X}^2=25-(1.4{)}^2$

$R{X}^2=23.04$

$RX=4.8$

$\therefore RM=2RX$

$=2\times 4.8$

$=9.6m$

$\therefore$ Distance between and Mandip in $9.6m$