In figure AB is a chord of a circle and At is a tangent at A such that angle BAT = 60° find
measure of angle acb
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120°
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Given: AB is chord of circle, AT is tangent to circle at A, angle BAT = 60°
To find: angle ACB
SOLUTION: As AT is tangent, and OA is radius of circle(as O is centre)
==> angle OAT = 90° [ Tangent of a circle is always perpendicular to the radius in point of contact]
As Angle BAT = 60°,
==> angle OAB= angle OAT - angle BAT
==> angle OAB= 90° - 60°
==> angle OAB = 30°
Now in ∆OAB, as OA= OB [ Radii of same circle],
==> angle OAB = angle OBA
==> angle OBA = 30°
Now in ∆OAB, using Angle Sum Property of ∆,
Angle OAB + angle OBA + angle AOB = 180°
==> 30°+30° + angle AOB = 180°
==> angle AOB = 180° - 30° -30°
==>angle AOB = 120°
Now, Reflex angle AOB = 360° - angle AOB [ Angles around a point add up to 360° ]
==> Reflex angle AOB = 360° - 120°
==> Reflex angle AOB = 240°
Thus, as angle subtended by an arc( major arc AB) at the centre is double the angle subtended by the same arc at the remaining part of the circle,
==> Reflex angle AOB = 2 × angle ACB
==> angle ACB = 1/2 × reflex angle AOB
==> angle ACB = 1/2 × 240°
==>angle ACB = 120°
THUS ANGLE ACB IS 120°
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