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Math Help - geometry (3)

  1. #1
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    geometry (3)

    In the diagram , AOB is a diameter of the circle with centre O . ACX , BDY and DEC are tangents to the circle at A , B and E respectively . Show that

    (1) angle COD =90

    (2) OC . OD = 1/2 AB . CD
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  2. #2
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    Hello thereddevils
    Quote Originally Posted by thereddevils View Post
    In the diagram , AOB is a diameter of the circle with centre O . ACX , BDY and DEC are tangents to the circle at A , B and E respectively . Show that

    (1) angle COD =90

    (2) OC . OD = 1/2 AB . CD
    Part (1) is like your previous question: Geometry (1). Here's the outline of the proof. I'll leave the details to you. Let \angle ACO = x, \angle EDO = y.

    Then \angle ECO = x and \angle BDO = y (congruent \triangle's)

    Then look at the quadrilateral ACDB. Can you see why 2x+2y = 180^o? Hence x+y=90^o. Then look at \triangle COD, and you're there.

    Part (2) Let AC = CE = p,\,ED=BD = q,\, AO=EO=BO =r.

    Then look at the areas:

    OC\cdot OD = 2\triangle OCD
    = 2(\triangle OCE + \triangle ODE)

    =2(\tfrac12pr + \tfrac12qr)

    =(p+q)r

    = \tfrac12CD\cdot AB
    Grandad
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