AB and ac are two chords of a circle with centre O as shown show that ao is bisector of ∠ BAC

Given: AB and AC are two equal chords of C (O, r).To prove: Centre, O lies on the bisector of ∠BAC.Construction: Join BC. Let the bisector of ∠BAC intersects BC in P.Proof:In ΔAPB and ΔAPC,AB = AC (Given)∠ BAP = ∠CAP (Given)AP = AP (Common)∴ Δ APB ≅ ΔAPC (SAS congruence criterion)⇒ BP = CP and ∠APB = ∠APC (CPCT)∠APB + ∠APC = 180° (Linear pair)⇒ 2 ∠APB = 180° (∠APB = ∠APC)⇒ ∠APB = 90°Now, BP = CP and ∠APB = 90°∴ AP is the perpendicular bisector of chord BC.

⇒ AP passes through the centre, O of the circle.

AB and AC are two equal chords of a circle. Prove that the bisector of the angle BAC passes through the centre of the circle. Can I prove that angle BOM=COM=90 degrees from maths

Question 4 Circles - Exercise 10.3

Answer:

According to the question,

We have,

AB and AC are two chords which are equal with centre O.

AM is the bisector of ∠BAC.

To prove: AM passes through O.

Construction: Join BC.

Let AM intersect BC at P.

Proof: In DBAP and DCAP

AB = AC [Given]

∠BAP = ∠CAP [Given]

And AP = BP [Common side]

△BAP ≅ △CAP [By SAS]

∠BPA = ∠CPA [CPCT]

We know that,

CP = PB

But, since ∠BPA and ∠CPA are linear pair angles,

We have,

∠BPA+∠CPA =180°

∠BPA = ∠CPA = 90°

Then,

AP is perpendicular bisector of the chord BC, which will pass through the centre O on being produced.

Therefore, AM passes through O.

Video transcript

"hello student welcome back to the leader doubt solving session so in today's session we are going to solve one more question related to the geometry all right so the question is a b and a c are the two equal chord of a circle the circle is given here all right and we are going to prove that the bisector of angle b a c passes through the center of the circle all right so according to the given question okay line segment or okay ray of the line okay that is a m okay bisect the angle bac angle [Music] b a c all right [Music] okay that means [Music] angle c a b c a b which is equals to angle [Music] vap all right being a p [Music] all right okay now here we are going to do one more construction here okay so what we are going to do okay we are going to join okay bc right so join the point join b c all right okay now we got the two triangles here right so we got the two triangle that means okay the first triangle is okay in triangle bap b a b and okay another triangle that we are going to consider that is c a b c a b right here ap is equals to ac right okay ap is equals to ac or it is ap is equals to ac okay so it is abc goes to ac okay which is given already so a b is equals to ac a b is equals to ac which is already given in the question only all right okay again as we know that okay angle okay b a p b a b is equals to angle c a b all right and again okay a b is equals to ap because it is a common side for the both the triangles all right it is a common side okay that means okay we can say that triangle b a b is concurrent to triangle c a p right both the triangles are concurrent triangles here all right okay because there are two sides and one angle is equal to each other all right okay that means here we can say that okay as both the triangles are concurrent we can say that okay cp is equals to bp all right because both the triangles are concurrent right and okay again okay now angle apc a p c plus angle bpa or it is apb angle apb is equals to 180 degree right because both the angles is a linear pair okay these are the linear square angle here angles okay that means okay the individual angle that is angle apc is equals to angle a b b which is equals to 90 degree all right okay and so that okay we can say that ap is a ap is perpendicular bisector of bisector of bc okay and which is passing through which is passing through passing through okay center of the circle center center of the circle all right that means okay we already proven that the angle uh we have already proven that the bisect bisector ap is passing through the center right that means it is proved all right so that is all about today's session if you are in need out here please do your please comment below in the comment section and subscribe to this channel so that you will get a notification for all upcoming videos thanks for watching this video bye "

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AB and AC are two equal chords of a circle. Prove that the bisector of the angle BAC passes through the centre of the circle.

Solution