NCERT Class 9 Solutions: Triangles (Chapter 7) Exercise 7.5 – Part 3

Intersection of perpendicular bisectors

Give intersection of perpendicular bisectors of triangle ABC at mid point of AB is M,mid point of AC is Q,mid point of BC is N

Give Intersection of Perpendicular Bisectors of Triangle ABC

Give intersection of perpendicular bisectors of triangle ABC at mid point of AB is M,mid point of AC is Q,mid point of BC is N

To find this point, you will construct three perpendicular bisectors, one for each side of the triangle. The point where all three perpendicular bisectors intersect is called the Circumcenter. Using this center point, we can draw a circle that passes through all three vertices.

Q-3 In a huge park, people are concentrated at three points (see figure):

  1. Where there are different slides and swings for children.

  2. Near which a man-made lack is situated.

  3. Which is near to a large parking and exit.

Point A A = (3.06, 2.52) Point A A = (3.06, 2.52) Point B B = (2.02, 0.62) Point B B = (2.02, 0.62) Point C C = (4.08, 0.36) Point C C = (4.08, 0.36) A text1 = "A" B text2 = "B" C text3 = "C"

Diagram Shows the People in the Park

Diagram shows people concentrated in the park (as a triangle)

Where an ice - cream parlor should be set up so that maximum number of persons can approach it?

Solution:

  • The parlor should be equidistant from A,B and C.

  • For this let we draw perpendicular bisector say l of line joining points B and C also draw perpendicular bisector say m of line joining points A and C.

Arc c Arc c: CircumcircularArc[M, N, O] Arc d Arc d: CircumcircularArc[Q, R, S] Arc e Arc e: CircumcircularArc[T, U, V] Arc r Arc r: CircumcircularArc[W, Z, A_1] Arc s Arc s: CircumcircularArc[B_1, I, C_1] Arc t Arc t: CircumcircularArc[D_1, E_1, F_1] Arc c_1 Arc c_1: CircumcircularArc[G_1, H_1, I_1] Arc d_1 Arc d_1: CircumcircularArc[J_1, K_1, L_1] Angle α Angle α: Angle between B, D, B' Segment f Segment f: Segment [A, B] Segment f Segment f: Segment [A, B] Segment g Segment g: Segment [B, C] Segment h Segment h: Segment [C, A] Segment i Segment i: Segment [C, D] Segment j Segment j: Segment [A, E] Segment k Segment k: Segment [E, B] Segment p Segment p: Segment [I, J] Segment q Segment q: Segment [K, L] A text1 = "A" B text2 = "B" C text3 = "C" P text4 = "P" Q text5 = "Q" R text6 = "R" L text7 = "L" M text8 = "M" O text9 = "O"

Triangle of ABC

Triangle of ABC l and m intersect each other at point O. Now point O is Equidistance from A,B and C. then join OA,OB,OC.

  • Let l and m intersect each other at point O.

  • Now point O is equidistant from points A, B and C.

  • Join OA, OB and OC.

  • Proof:

    In BOP and COP ,

    OP=OP [Common]

OPB=OPC=90°

BP=PC [P is the mid-point of BC]

BOPCOP [By Side-Angle-Side congruency]

OB=OC [By Corresponding Parts Congruent Triangles] …..equation (1)

  • Similarly, AOQCOQ

    OA=OC [By Corresponding Parts Congruent Triangles] …..equation (2)

  • From eq. (1) and (2),

    OA=OB=OC

  • Therefore, ice cream parlour should be set up at point O, the point of intersection of perpendicular bisectors of any two sides out of three formed by joining these points.

Explore Solutions for Mathematics

Sign In