A, B and C are three points on a circle. Prove that the perpendicular bisectors of AB, BC and CA are concurrent.
Solution:
Given, A, B and C are three points on a circle.
We have to prove that the perpendicular bisectors of AB, BC and CA are concurrent.
Draw perpendicular bisectors of AB and AC which meet at a point O.
Join OA, OB and OC.
Perpendicular bisector of BC also passes through O
So, OL = ON = OM
Considering triangles OEA and OEB,
Since OL is the perpendicular bisector of AB
AE = BE
∠AEO = ∠BEO = 90°
Common side = OE
SAS criterion states that if two sides of one triangle are proportional to two sides of another triangle and their included angles are congruent, then the triangles are similar.
By SAS criterion, the triangles OEA and OEB are similar.
So, OA = OB
Similarly, by SAS criterion, the triangles OFA and OFC are similar.
By CPCTC,
OA = OC
Let OA = OB = OC = r
Draw a perpendicular from O to BC and join them.
Considering triangles OMB and OMC,
OB = OC
Common side = OM
∠OMB = ∠OMC = 90°
RHS criterion states that if the hypotenuse and side of one right-angled triangle are equal to the hypotenuse and the corresponding side of another right-angled triangle, the two triangles are congruent.
By RHS criterion, the triangles OMB and OMC are similar.
The Corresponding Parts of Congruent Triangles are Congruent (CPCTC) theorem states that when two triangles are similar, then their corresponding sides and angles are also congruent or equal in measurements.
By CPCTC,
BM = MC
This implies OM is the perpendicular bisector of BC
Therefore, OL, OM and ON are concurrent.
✦ Try This: Two congruent circles intersect each other at points A and B. Through A any line segment PAQ is drawn so that P,Q lie on the two circles. Prove that BP = BQ.
☛ Also Check: NCERT Solutions for Class 9 Maths Chapter 10
NCERT Exemplar Class 9 Maths Exercise 10.3 Problem 3
A, B and C are three points on a circle. Prove that the perpendicular bisectors of AB, BC and CA are concurrent.
Summary:
A, B and C are three points on a circle. It is proven that the perpendicular bisectors of AB, BC and CA are concurrent
☛ Related Questions:
- AB and AC are two equal chords of a circle. Prove that the bisector of the angle BAC passes through . . . .
- If a line segment joining mid-points of two chords of a circle passes through the centre of the circ . . . .
- ABCD is such a quadrilateral that A is the centre of the circle passing through B, C and D. Prove th . . . .