Home
Class 9
MATHS
Two chords A B and AC of a circle are eq...

Two chords `A B` and `AC` of a circle are equal. Prove that the centre of the circle lies on the angle bisector of `/_B A Cdot`

Text Solution

Verified by Experts

Given : `AB` and `AC` are two equal chords of a circle `C (O,r)` and `AD` is the bisector of `/_ BAC `.
To Prove : lies on `AD`
Construction : Join `BC`, meeting `AD` at `M`.
Proof : In `Delta` BAM and `Delta CAM`, we have
`AB = AC` [given]
`/_ BAM= /_ CAM` [given]
`AM = AM` [common]
`:. Delta BAM ~=Delta CAM `[by SAS-congurence]
`implies BM =CM` and `/_BMA=/_CMA ` [c.p.c.t.]
`implies BM=CM` and `/_BMA /_CMA ` and `/_BMA =/_ CMA implies /_BMA =/_CMA =90^(@)]`
`implies `AM is the perpendicular bisector of chord BC

`implies ` `AD` is the perpendicular bisector of the chord `BC`.
But, the perpendicular bisector of a chord always passes through the centre of the circle.
`:. ` `AD` passes through the centre `O` of the circle.
`implies` `O` lies on `AD`.
Promotional Banner

Similar Questions

Explore conceptually related problems

Two chords AB and AC of a circle are equal. Prove that the centre of the circle lies on the angle bisector of /_BAC.

Two chord AB and AC of a circle are equal. Prove that the centre of a circle lies on the angle bisector of /_BAC

If two chords AB and AB of a circle with centre O are such that the centre O lies on the bisector of /_BAC; Prove that AB=AC

If two chords AB and AC of a circle with centre O are such that the centre O lies on the bisector of /_BAC, prove that AB=AC, i.e. the chords are equal.

Equal chords of a circle subtend equal angles at the centre.

Theorem: 4( iii) If two equal chords are drawn from a point on the circle; then the centre of a circle will lie on angle bisector of these two chords.

If the length of a chord of a circle is equal to that of the radius of the circle , then the angle subtended , in radians , at the centre of the circle by the chord is

Prove that the centre of a circle touching two intersecting lines lies on the angle bisector of the lines.

Theorem 10.1 : Equal chords of a circle subtend equal angles at the centre

In the given figure,AB and AC are two equal chords of a circle with centre O.Show that o lies on the bisectors of /_BAC.