Results 1 to 9 of 9

Math Help - Radius-point of tangency proof

  1. #1
    Newbie
    Joined
    Apr 2010
    Posts
    6

    Radius-point of tangency proof

    Also, can anyone prove that the radius which shares the same point of tangency is perpendicular to the tangent line.

    "A tangent is perpendicular to the radius that shares the point of tangency."

    Thanks alot!
    Follow Math Help Forum on Facebook and Google+

  2. #2
    Lord of certain Rings
    Isomorphism's Avatar
    Joined
    Dec 2007
    From
    IISc, Bangalore
    Posts
    1,465
    Thanks
    6
    Quote Originally Posted by lpbug View Post
    Also, can anyone prove that the radius which shares the same point of tangency is perpendicular to the tangent line.

    "A tangent is perpendicular to the radius that shares the point of tangency."

    Thanks alot!
    Take a paper and draw the following:

    Let the centre of the circle be O. Let PQ be a tangent touching the circle at T. Clearly OT is a radial line (radius).
    If OT was not perpendicular to PQ, then we can drop a perpendicular from O onto PQ and call the foot of the perpendicular S. Thus we see that OS (the perpendicular) is shorter than OT (some other line joining O and a point on PQ). But is that possible?
    Follow Math Help Forum on Facebook and Google+

  3. #3
    Newbie
    Joined
    Apr 2010
    Posts
    6
    I don't really get it... I mean, why would OS be shorter than OT? it would be longer...
    Follow Math Help Forum on Facebook and Google+

  4. #4
    Lord of certain Rings
    Isomorphism's Avatar
    Joined
    Dec 2007
    From
    IISc, Bangalore
    Posts
    1,465
    Thanks
    6
    Quote Originally Posted by lpbug View Post
    I don't really get it... I mean, why would OS be shorter than OT? it would be longer...
    Exactly!!
    If OT was not perpendicular to PQ, OS is shorter than OT. But that cannot happen! Thus OT is perpendicular to PQ
    Follow Math Help Forum on Facebook and Google+

  5. #5
    Newbie
    Joined
    Apr 2010
    Posts
    6
    OH wait, so it would mean that a perpendicualr dropped from O onto any of the lines besides a tangent line would mean that it would be shorter than the radius?
    Follow Math Help Forum on Facebook and Google+

  6. #6
    Lord of certain Rings
    Isomorphism's Avatar
    Joined
    Dec 2007
    From
    IISc, Bangalore
    Posts
    1,465
    Thanks
    6
    Quote Originally Posted by lpbug View Post
    OH wait, so it would mean that a perpendicualr dropped from O onto any of the lines besides a tangent line would mean that it would be shorter than the radius?
    Thats a well known result. It has nothing to do with tangents. Pick any line AB and a point M not on the line. The shortest line joining M to a point on AB is a line perpendicular to AB and passing through M.
    Follow Math Help Forum on Facebook and Google+

  7. #7
    Newbie
    Joined
    Apr 2010
    Posts
    6
    hmm, i do see your point now, but still, are there any way to proove this by using a two colum proof? like using numbers and stuff to prove it?
    Follow Math Help Forum on Facebook and Google+

  8. #8
    Super Member
    Joined
    Jun 2009
    Posts
    806
    Thanks
    4
    Quote Originally Posted by lpbug View Post
    hmm, i do see your point now, but still, are there any way to proove this by using a two colum proof? like using numbers and stuff to prove it?
    There is one theorem in circle.
    If you draw a chord AB from the point of contact (A) of the tangent to the circle, then the angle subtended by the chord at any point P on the circumference (angle APB ) is equal to the angle between chord and the tangent.
    Now suppose the chord AB is the diameter of the circle. Then angle APB = π/2. So the angle between AB ( i.e. radius ) and the tangent is π/2
    Follow Math Help Forum on Facebook and Google+

  9. #9
    No one in Particular VonNemo19's Avatar
    Joined
    Apr 2009
    From
    Detroit, MI
    Posts
    1,823
    Calclus makes this really easy.

    A line from the center (x_0,y_0) to a point (x,y) on the circoe Has equation y-y_0=\frac{y-y_0}{x-x_0}(x-x_0). Differentiating the equatio of a circle we get : \frac{dy}{dx}= -\frac{x-x_0}{y-y_0}.
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. Tangency of circles - proof
    Posted in the Geometry Forum
    Replies: 1
    Last Post: October 12th 2011, 11:49 AM
  2. radius given a point on circumference
    Posted in the Geometry Forum
    Replies: 4
    Last Post: September 1st 2010, 04:12 PM
  3. Finding radius from just a point
    Posted in the Trigonometry Forum
    Replies: 2
    Last Post: March 22nd 2010, 07:14 PM
  4. Circles question (point of tangency stuff)
    Posted in the Trigonometry Forum
    Replies: 1
    Last Post: December 9th 2008, 04:17 AM
  5. Finding integers? Point of tangency?
    Posted in the Calculus Forum
    Replies: 5
    Last Post: August 22nd 2008, 01:52 PM

Search Tags


/mathhelpforum @mathhelpforum