In fig., l and m are two parallel tangents to a circle with centre O,touching the circle at A and B respectively. Another tangent at C intersects the line l at D and m at E. Prove that $∠$ DOE = 90⁰

Solution

$Given : l and m are two parallel tangents to the circle with centre O touching the circle at A and B respectively . DE is a tangent at the point C, which intersect l at D and m at E . To prove : ∠ DOE = 90 ° Contruction : Join OC . Proof :$

$In ∆ ODA and ∆ ODC, OA = OC (Radii of the same circle) AD = DC (Length of tangents drawn from an external point to a circle are equal) DO = OD (Commmon side) ∴ ∆ ODA ≅ ∆ ODC, (SSS congruence criterion) ∴ ∠ DOA = ∠ COD . . . . . . . . . . (1) Similarly, ∆ OEB ≅ ∆ OEC ∴ ∠ EOB = ∠ COE . . . . . . . . . . . (2) Now, AOB is a diameter of the circle . Hence, it is a straight line . ∠ DOA + ∠ COD + ∠ COE + ∠ EOB = 180 ° From (1) and (2), we have : \Rightarrow 2 ∠ COD + 2 ∠ COE = 180 ° \Rightarrow ∠ COD + ∠ COE = 90 ° \Rightarrow ∠ DOE = 90 ° Hence, proved .$

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In fig., l and m are two parallel tangents to a circle with centre O,touching the circle at A and B respectively. Another tangent at C intersects the line l at D and m at E. Prove that $∠$ DOE = 90⁰

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In fig., l and m are two parallel tangents to a circle with centre O,touching the circle at A and B respectively. Another tangent at C intersects the line l at D and m at E. Prove that $∠$ DOE = 90⁰