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Math Help - a question on properties of circle

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    a question on properties of circle

    Referring to the attached pic, work out the length of OC - do not use trigonometric ratios stuff if possible.
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    Draw a tangent to the circle from C, say CT is the tangent and OT is the radius (so the lines CT and OT are perpendicular). There's a theorem which says that CA\times CB = CT^2, from which you can find CT. Then use Pythagoras.
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    Quote Originally Posted by Opalg View Post
    Draw a tangent to the circle from C, say CT is the tangent and OT is the radius (so the lines CT and OT are perpendicular). There's a theorem which says that CA\times CB = CT^2, from which you can find CT. Then use Pythagoras.
    produce 2 similar triangles?
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    Quote Originally Posted by Opalg View Post
    Draw a tangent to the circle from C, say CT is the tangent and OT is the radius (so the lines CT and OT are perpendicular). There's a theorem which says that CA\times CB = CT^2, from which you can find CT. Then use Pythagoras.
    but how do you know that TC is certainly a tangent of the circle???
    actually if TC is produced from the original figure, angle ATC is does not appear to be a right angle.
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    Quote Originally Posted by ukorov View Post
    but how do you know that TC is certainly a tangent of the circle???
    actually if TC is produced from the original figure, angle ATC is does not appear to be a right angle.
    This is the picture that I intended. The angle OTC is a right angle.
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    Quote Originally Posted by Opalg View Post
    This is the picture that I intended. The angle OTC is a right angle.
    oh i see but why is CA x CB = CT^2 ?
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    Quote Originally Posted by ukorov View Post
    Referring to the attached pic, work out the length of OC - do not use trigonometric ratios stuff if possible.
    Is it considered to be using trigonometric ratios
    if you find the mid-point of the chord AB?

    Since the chord length is given as 8, half would be 4.
    If not then you have two 3-4-5 triangles inside the circle.
    You then have sufficient information to complete the sides of a right triangle.
    And from there, use pythagoras.

    .
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    Quote Originally Posted by aidan View Post
    Is it considered to be using trigonometric ratios
    if you find the mid-point of the chord AB?

    Since the chord length is given as 8, half would be 4.
    If not then you have two 3-4-5 triangles inside the circle.
    You then have sufficient information to complete the sides of a right triangle.
    And from there, use pythagoras.

    .
    well it is pretty easy to solve using tri ratios but this question was extracted from a textbook chapter on properties of circle.
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    Quote Originally Posted by ukorov View Post
    oh i see but why is CA x CB = CT^2 ?
    Similar triangles. The triangles CTB and CAT are similar (angles are equal by the alternate segment theorem). Therefore \frac{CA}{CT} = \frac{CT}{CB}.
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