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In the Given Figure, O is the Centre of the Circumcircle Abc. Tangents at a and C Intersect at P. Given Angle Aob = 140° and Angle Apc = 80°; Find the Angle Bac. - ICSE Class 10 - Mathematics

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ConceptTangent Properties - If a Line Touches a Circle and from the Point of Contact, a Chord is Drawn, the Angles Between the Tangent and the Chord Are Respectively Equal to the Angles in the Corresponding Alternate Segments

Question

In the given figure, O is the centre of the circumcircle ABC. Tangents at A and C intersect at P. Given angle AOB = 140° and angle APC = 80°; find the angle BAC.

Solution

 

Join OC.
Therefore, PA and PA are the tangents
∴ OA  ⊥ PA and OC ⊥  PC
In quadrilateral APCO,
`∠`APC+ `∠` AOC  =  180°
⇒ 80° + `∠`AOC =  180°
⇒ `∠`AOC = 100° 
`∠` BOC = 360° - ( `∠` AOB + `∠`AOC)
`∠` BOC ≅  360°  - (140° + 100° )
`∠`B = 360° - 240° =120°
Now, arc BC subtends  `∠`BOC at the centre and `∠`BAC at the remaining part of the circle
 ∴ `∠`BAC = `1/2` `∠`BOC
 `∠`BAC `=1/2`  × 120° = 60°


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Solution In the Given Figure, O is the Centre of the Circumcircle Abc. Tangents at a and C Intersect at P. Given Angle Aob = 140° and Angle Apc = 80°; Find the Angle Bac. Concept: Tangent Properties - If a Line Touches a Circle and from the Point of Contact, a Chord is Drawn, the Angles Between the Tangent and the Chord Are Respectively Equal to the Angles in the Corresponding Alternate Segments.
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