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In the Given Figure Pq is a Tangent to the Circle at A, Ab and Ad Are Bisectors of `Anglecaq` and `Anglepac`. If `Anglebaq = 30^@. Prove That:Bd is a Diameter of the Circle and Abc is an Isosceles Triangle - ICSE Class 10 - Mathematics

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Question

In the given figure PQ is a tangent to the circle at A, AB and AD are bisectors of `angleCAQ` and `angle PAC`. if `angleBAQ = 30^@. prove that:

1) BD is a diameter of the circle

2) ABC is an isosceles triangle

Solution

1) `angleBAQ = 30^@`

SinceAB is the bisector of `angleCAQ`

`=> angleCAB = angleBAQ = 30^@`

AD is the bisector of `angle CAP` and P-A-Q,

`angle DAP + angle CAD + angle CAQ = 180^@`

`=> angleCAD + anglle CAD + 60^@ = 180 ^@`

`=> angle CAD = 60^@`

So `angle CAD + angle CAB = 60^@ + 30^@ = 90^@`

Since angle in a semi-circle = 90°

⇒ Angle made by diameter to any point on the circle is 90°

So, BD is the diameter of the circle.

2) SinceBD is the diameter of the circle, so it will pass through the centre.

By Alternate segment theorem

`angle ABD = angle DAC = 60^@`

So, in `angle BMA`,

`angle AMB = 90^@`    .........(UseAngleSumProperty)

We know that perpendicular drawn from the centre to a chord of a circle bisects the chord.

`=> angle BMA = angle BMC = 90^@`

In `triangleBMA` and `triangleBMC` 

`angleBMA = angleBMC = 90^@`

BM = BM (common side)

AM = CM(perpendicular drawn from the centre to a chord of a circle bisects the chord.)

⇒ ΔBMA ≅ ΔBMC

⇒ AB = BC (SAS congruence criterion)

⇒ ΔABC is an isosceles triangle.

  Is there an error in this question or solution?

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Solution In the Given Figure Pq is a Tangent to the Circle at A, Ab and Ad Are Bisectors of `Anglecaq` and `Anglepac`. If `Anglebaq = 30^@. Prove That:Bd is a Diameter of the Circle and Abc is an Isosceles Triangle Concept: Construction of Tangents to a Circle.
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