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In the figure, O is the centre of the ci...

In the figure, O is the centre of the circle . Seg AB is the diameter at the point C on the circle the tangent CD is drawn. Line BD is a tangent to the circle at point B. Show that seg OD `||` chord AC.

Text Solution

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In `Delta OAC`,
seg `OA ~= ` seg OC ….(Radii of the same circle )
`:. /_ OAC ~= /_ OCA ` ….( Isosceles triangle theorem ) ….(1)
`/_COB` is an exterior angle of `Delta OAC`,
`/_ COB = /_ OAC + /_ OCA` ….(Remote interior angles theorem )
`:. /_ COB = /_ OAC + /_ OAC ` ....[From (1) ] ,brgt `:. /_ COB = 2/- OAC ` ...(2)
In `Delta OCD ` and `Delta OBD`,
seg `OC ~= ` seg OB ....( Radii of same circle )
seg OD `~=` seg OB ....(Common side )
seg `DC~= `seg DB ....(Tangent segment theorem )
`:. Delta OCD ~= Delta OBD ` .....(SSS test of congruence )
`:/_ COD ~= /_ BOD ` ...(c.a.c.t. ) ...(3)
`/_ COB = /_ COD + /_ BOD ` ....( Angle addition axiom )
`:. /_COB = /_ BOD + /_BOD ` .....[From (3) ]
`:. /_COB = 2/_ BOD ` ...(4)
`:.` from (20 and (4)
`2/_OAC = 2 /_ BOD `
`:. /_OAC = /_ BOD `
`:.` seg `OD ||` chord AC ....(Corresponding angles test )
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