Q. 34.2( 306 Votes )

# If two circles in

Answer :

Let two circles with centres O and O' respectively intersect at two points A and B
such that AB is the common chord of two circles and
OO' is the line segment joining the centres, as shown in the figure: Also, AB is the chord of the circle centered at O. Therefore, the perpendicular bisector of AB will pass through O.

Let OO' intersect AB at M. Also, draw line segments OA, OB, O'A and O'B. Now, In ΔOAO' and OBO',
OA = OB (radii of same circle)
O'A = O'B (radii of same circle)
O'O ( common side)
⇒ ΔOAO' ≅ ΔOBO' (Side-Side-Side congruency)
⇒ ∠AOO' = ∠BOO'
⇒ ∠AOM = ∠BOM ......(i)

Now in ΔAOM and ΔBOM we have
OA = OB (radii of same circle)
∠AOM = ∠BOM (from (i))
OM = OM (common side)
⇒ ΔAOM ≅ ΔBOM (Side-Angle-Side congruency)
⇒ AM = BM and ∠AMO = ∠BMO
But
∠AMO + ∠BMO = 180°
⇒ 2∠AMO = 180°
⇒ ∠AMO = 90°
Thus, AM = BM and ∠AMO = ∠BMO = 90°
Hence OO' is the perpendicular bisector of AB.

Rate this question :

How useful is this solution?
We strive to provide quality solutions. Please rate us to serve you better.
Related Videos  IMP Qs Related with Circles For Boards44 mins  Arc of Circle And Related IMP Qs40 mins  Chords Of Circles42 mins  IMP Theorems And Their Application43 mins  Circles- All kinds of Questions36 mins
Try our Mini CourseMaster Important Topics in 7 DaysLearn from IITians, NITians, Doctors & Academic Experts
Dedicated counsellor for each student
24X7 Doubt Resolution
Daily Report Card
Detailed Performance Evaluation view all courses RELATED QUESTIONS :

PQ and RQ are theRS Aggarwal & V Aggarwal - Mathematics

In a circle of raRD Sharma - Mathematics

Two equal circlesRD Sharma - Mathematics

Angle formed in mRD Sharma - Mathematics

If AB is aRD Sharma - Mathematics

The greatest chorRD Sharma - Mathematics

If two diameters RD Sharma - Mathematics

<span lang="EN-USRS Aggarwal & V Aggarwal - Mathematics