NCERT Class 9 Solutions: Circles (Chapter 10) Exercise 10.4 – Part 3

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.

Drawing circumcenter of triangle ABC

Circumcenter of Triangle ABC

Drawing circumcenter of triangle ABC

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:

Circle c Circle c: Circle through R with center O Angle ? Angle ?: Angle between S, M, D' Angle ? Angle ?: Angle between S, M, D' Segment f Segment f: Segment [P, R] Segment g Segment g: Segment [S, O] Segment h Segment h: Segment [S, P] Segment i Segment i: Segment [S, R] Segment j Segment j: Segment [P, O] Point O O = (-0.16, 2.3) Point O O = (-0.16, 2.3) Point O O = (-0.16, 2.3) Point R R = (1.6, 3.68) Point R R = (1.6, 3.68) Point R R = (1.6, 3.68) Point P Point P: Point on c Point P Point P: Point on c Point P Point P: Point on c Point S Point S: Point on c Point S Point S: Point on c Point S Point S: Point on c Point M Point M: Intersection point of f, g Point M Point M: Intersection point of f, g Point M Point M: Intersection point of f, g

P, S, R Represent on Circle

P, S, R represented on circle, O is a centre point, also drawn SM ⊥ PR

  • Lets consider Reshma, Salma and Mandip to be standing at P,S and R respectively.

  • We know that, PS=6m and SR=6m.

  • Radius OP=5m

  • SMPR is drawn.

  • Now, PSR is an isosceles triangle as PS=SR , 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 PM=y and OM=x then SM=(5x) . (radius OS=5m )

Applying Pythagoras theorem in ΔOPM ,

  • OP2=OM2+PM2

  • 52=x2+y2 …………equation (1)

Applying Pythagoras theorem in ΔPMS ,

  • PS2=SM2+PM2

  • 62=(5x)2+y2 ……..equation (2)

Subtracting (1) from (2), we get

  • 6252=(5x)2+y2(x2+y2)

  • 3625=2510x+x2+y2x2y2

  • 11=2510x

  • 10x=14

  • x=1410

  • x=75

Put the value of x in (1),

  • 52=x2+y2

  • 25=(4925)+y2

  • y2=254925

  • y2=6254925

  • y2=57625

  • y=245 ( 576=24,25=5 )

So,

  • PR=2×PM = 2×y=2×(24/5)m = 48/5m =9.6m

  • Distance between Reshma and Mandip is 9.6m .

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:

Circle c Circle c: Circle through R with center O Segment f Segment f: Segment [Q, R] Segment g Segment g: Segment [Q, P] Segment h Segment h: Segment [P, R] Segment i Segment i: Segment [P, S] Point O O = (-0.1, 2.66) Point O O = (-0.1, 2.66) Point O O = (-0.1, 2.66) Point R R = (2.16, 1.56) Point R R = (2.16, 1.56) Point R R = (2.16, 1.56) Point Q Point Q: Point on c Point Q Point Q: Point on c Point Q Point Q: Point on c Point P Point P: Point on c Point P Point P: Point on c Point P Point P: Point on c Point S Point S: Point on f Point S Point S: Point on f Point S Point S: Point on f

Circle Point Is P, Q, and R

Circle with points at P, Q and R, also PS ⊥ QR, PS is median of ΔPQR

  • 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 PSQR , then following properties will hold:

  • Because ΔPQR is an equilateral triangle, perpendicular bisector of a side from opposite vertex is also the median. Therefore, PS will be median of ΔPQR , 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, OP=23PS

  • Let the side of a triangle QR is am then QS=a2m.

Applying Pythagoras theorem in ΔPQS ,

  • PQ2=QS2+PS2

  • PS2=PQ2QS2

  • PS2=a2(a2)2

  • PS2=a2a24

  • PS2=3a24

  • PS=3a22

  • OP=23PS

  • 20m=23×3a2 ( OP=20m )

  • 20m=3a3

  • 20m=13a

  • a=203m

Therefore, length of the string is 203m .

Explore Solutions for Mathematics

Sign In