Results 1 to 2 of 2

Math Help - Stuck on proof of Secant-Tangent Theorem

  1. #1
    Member ilikedmath's Avatar
    Joined
    Sep 2008
    Posts
    98

    Exclamation Stuck on proof of Secant-Tangent Theorem

    Secant-Tangent Theorem states: If a secant PA and tangent PC meet a circle at the respective points A, B, and C (point of contact),
    then (PC)^2 = (PA)(PB).




    My work so far on the proof:
    Given circle O with secant PA and tangent PC which meet circle O at A, B, and C.
    Draw chords AC and BC.
    Angle CPA is congruent to Angle CPB.
    Angle PCA is congruent to Angle PBC (both angles intercept arc AC).
    Triangle ACP ~ Triangle DBP by the Angle-Angle Similarity Criterion.

    Now, this is where I am stuck. I know the two triangles are similar. My hunch is to manipulate the relations of corresponding parts of similar triangles to get to what I want to prove that
    (PC)^2 = (PA)(PB).

    Will the Inscribed Angle Theorem help any?
    (Inscribed Angle Theorem: The measure of an inscribed angel of a circle equals 1/2 that of its intercepted arc.)

    Any help is greatly appreciated! Thank you for your time.



    Follow Math Help Forum on Facebook and Google+

  2. #2
    MHF Contributor red_dog's Avatar
    Joined
    Jun 2007
    From
    Medgidia, Romania
    Posts
    1,252
    Thanks
    5
    \Delta ACP\sim\Delta CBP\Rightarrow \frac{PC}{PB}=\frac{PA}{PC}\Rightarrow PC^2=PA\cdot PB
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. Tangent and Secant line
    Posted in the Calculus Forum
    Replies: 1
    Last Post: January 2nd 2012, 12:53 PM
  2. Replies: 8
    Last Post: February 20th 2010, 07:52 PM
  3. tangent/secant lines.
    Posted in the Calculus Forum
    Replies: 5
    Last Post: October 7th 2009, 06:44 PM
  4. Trig Integration -- secant and tangent problem
    Posted in the Calculus Forum
    Replies: 2
    Last Post: September 27th 2009, 03:26 PM
  5. tangent and secant
    Posted in the Calculus Forum
    Replies: 3
    Last Post: March 29th 2008, 05:24 AM

Search Tags


/mathhelpforum @mathhelpforum