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AP is tangent to circle O at point P, Wh...

AP is tangent to circle O at point P, What is the length of OP?

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Let the radius of given circle is r.
`:.` `OP=OB=r`
`:.` `OA=2+r,` `OP=r,` `AP=4`
`:." "OP=OB=r`
`:." "OA=2+r," "OP=r," "AP=4`
`angleOPA=90^(@)" "`(radius through point of contact is perpendicular to the tangent)

`:.` In right `triangleOPA,`
`OA^(2)=OP^(2)+AP^(2)" "` (by Pythagoras theorem)
`(2+r^(2))=r^(2)+(4)^(2)`
`implies" "4+r^(2)+4r=r^(2)+16`
`(2+r^(2))=r^(2)+(4)^(2)`
`implies" "4+r^(2)+4r=r^(2)+16`
`implies" "4r=12" "implies" "r=3`
`:." "OP=3 cm.`
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