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In the given figure, AB is the diameter of the circle, with center O, and AT is the tangent. Calculate the numerical value of x.

In the figure, AB is the diameter of the circle, with center O, and AT is the tangent. Calculate the numerical value of x. Tangents and Intersecting Chords, Concise Mathematics Solutions ICSE Class 10.

Circles

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Answer

In △OBC,

In the figure, AB is the diameter of the circle, with center O, and AT is the tangent. Calculate the numerical value of x. Tangents and Intersecting Chords, Concise Mathematics Solutions ICSE Class 10.

OB = OC (Radius of same circle)

As, angles opposite to equal sides are equal.

∴ ∠OBC = ∠OCB

As, exterior angle is equal to the sum of two opposite interior angles.

∴ ∠COA = ∠OBC + ∠OCB

⇒ ∠COA = 2∠OBC

⇒ 2∠OBC = 64°

⇒ ∠OBC = 32°.

In △ABT,

∠BAT = 90° (∵ Tangent at any point of a circle and the radius through this point are perpendicular to each other.)

⇒ ∠BAT + ∠ABT + ∠ATB = 180° (By angle sum property of triangle)

⇒ 90° + 32° + x° = 180° [∵ ∠ABT and ∠OBC is the same angle]

⇒ x° + 122° = 180°

⇒ x° = 180° - 122°

⇒ x° = 58°.

Hence, x = 58°.

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