# NCERT Class 9 Solutions: Circles (Chapter 10) Exercise 10.4 – Part 3 (For CBSE, ICSE, IAS, NET, NRA 2022)

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Pythagoras theorem

The square of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the other two sides.

Q-5 Three girls Reshma, Salma and Mandip are playing a game by standing on a circle of radius 5m 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 6m each, what is the distance between Reshma and Mandip?

Solution:

- Lets consider Reshma, Salma and Mandip to be standing at P, S and R respectively.
- We know that, and
- Radius
- is drawn.
- Now, PSR is an isosceles triangle as , M is mid-point of PR.
- Since SM is perpendicular bisector of PR (chord) therefore, it passes through the centre of the circle (perpendicular bisector of chord of a circle passes through the center) .
- Let and then . (radius )

Applying Pythagoras theorem in ,

- … equation (1)

Applying Pythagoras theorem in ,

- … equation (2)

Subtracting (1) from (2) , we get

Put the value of in (1) ,

- ()

So,

- =
- Distance between Reshma and Mandip is .

Q-6 A circular park of radius 20m is situated in a colony. Three boys Ankur, Syed and David are sitting at equal distance on its boundary each having a toy telephone in his hands to talk each other. Find the length of the string of each phone.

Solution:

- Let consider Ankur, Syed and David to be at P, Q and R respectively.
- All three boys are at equal distances thus PQR is an equilateral triangle.
- Since QR is the chord of the circle if we draw a line PS such that , then following properties will hold:
- Because is an equilateral triangle, perpendicular bisector of a side from opposite vertex is also the median. Therefore, PS will be median of , which means it will bisect QR.
- Since PS is perpendicular bisector of a chord (QR) therefore it passes through the centre O.
- The incircle of the triangle PQR is at O. Therefore OP is the radius of the triangle is OP
- Now since PS is the median of the triangle,
- Let the side of a triangle QR is then

Applying Pythagoras theorem in ,

- ()

Therefore, length of the string is .