# Ex.10.4 Q5 Circles Solution - NCERT Maths Class 9

## Question

Three girls Reshma, Salma and Mandip are playing a game by standing on a circle of radius \(5\rm\,m\) drawn 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 \(6\,\rm m\) each, what is the distance between Reshma and Mandip?

## Text Solution

**What is known?**

Three players are standing in a circle. Distance between two pairs is given. Radius of circle is given.

**What is unknown?**

Distance between Reshma and Mandip

**Reasoning:**

Perpendicular from center to either of the chord bisects the chord.Using this fact and by pythagoras theorem we can find value of \(OA\). After that we can find area of \(∆ORS\) by two ways like \(RS\) as base and \(OA\) as height or \(OS\) as base and \(RN\) as height. From this we will get the value of \(RN\) and double of value \(RN\) will will give the distance between Reshma and Mandip.

Steps:

Let \(O\) be the centre of the circle and \(R, M\) and \(S\) denote Reshma, Mandip and Salma respectively.

Draw a perpendicular \({OA}\) to \({RS}\) from \({O.}\) Then \(\begin{align}{RA = AS = 3}\,\rm {m.} \end{align}\)

Using Pythagoras theorem we get, \(\begin{align}OA = 4 \text{m.} \end{align}\)

We can see that quadrilateral \(ORSM\) takes that shape of a kite.

(Because \(OR = OM\) and \(RS = SM\)).

We know that the diagonals of a kite are perpendicular and the main diagonal bisects the other diagonal.

\(\angle {RNS}\) will be \(90^\circ\) and \(\begin{align}{RN = NM} \end{align}\)

\(\text{Area of } ∆ORS\) \[\begin{align} &= \frac{1}{2} \times RS \times OA\\ &=\frac{1}{2} \times 6 \times4\\&= 12 \dots(1)\end{align}\]

Also

\(\text{Area of} ∆ORS\) \[\begin{align}& =\frac{1}{2} \times OS \times RN\\&= \frac{1}{2} \times 5 \times RN\dots(2)\end{align}\]

From equation (\(1\)) and (\(2\))

\[\begin{align}\frac{1}{2} \times 5 \times RN\,= 12\end{align}\]

\[\begin{align}RN = \frac{{24}}{5} = 4.8\, \rm m\end{align}\]

\[\begin{align}RM = 2 × RN = 2 × 4.8 = 9.6 \, \rm m\end{align}\]

Distance between Reshma and Salma is \(9.6\, \rm{ m.}\)