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Abcd is a Cyclic Quadrilateral with Bc = Cd. Tc is Tangent to the Circle at Point C and Dc is Produced to Point G. If ∠Bcg = 108° and O is the Centre of the Circle Find: (I) Angle Bct (Ii) Angle Doc - ICSE Class 10 - Mathematics

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Question

In the figure, ABCD is a cyclic quadrilateral with BC = CD. TC is tangent to the circle at point C and DC is produced to point G. If ∠BCG = 108° and O is the centre of the circle, find:
(i) Angle BCT
(ii) angle DOC

Solution

Join OC, OD and AC
i)
`∠`BCG + `∠`BCD = 180° (Linear pair)
⇒ 180 °+  `∠`BCD  = 180°
⇒ `∠`BCD = 180°  -180°  =72°
BC = CD
∴ `∠`DCP = `∠`BCT
But, `∠`BCT + `∠`BCD + `∠`DCP = 180°
∴ `∠`BCT  + `∠`BCT + 72° = 180°
2`∠`BCT = 180°  - 72°

`∠`BCT = 54°

ii)
PCT is a tangent and CA is a chord.
`∠`CAD  = `∠`BCT = 54°
But arc DC subtends `∠`DOC at the centre and `∠`CAD at the
remaining part of the circle.
∴ `∠`DOC = 2`∠`CAD = 2 × 54° = 108°

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Solution Abcd is a Cyclic Quadrilateral with Bc = Cd. Tc is Tangent to the Circle at Point C and Dc is Produced to Point G. If ∠Bcg = 108° and O is the Centre of the Circle Find: (I) Angle Bct (Ii) Angle Doc Concept: Cyclic Properties.
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