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Tangents to Circles


Date: 06/25/99 at 10:34:23
From: Thaddeus
Subject: Tangents to circles

I want to try to prove that a line L is tangent to Circle C if and 
only if L is perpendicular to ZA, where Z is the center of my circle 
and A is a point on circle C. It gave me the hint to consider the 
Pythagorean theorem, but I am unsure of how to begin.


Date: 06/25/99 at 12:41:53
From: Doctor Rob
Subject: Re: Tangents to circles

Thanks for writing to Ask Dr. Math!

You didn't say this, but you need A to lie on L for this to be true.

Show that if L is perpendicular to ZA at A, then every point on L 
other than A is farther from Z than A is, and so lies outside the 
circle. Let P be such a point. Then AZP is a right triangle, and you 
can use the Pythagorean Theorem to show that AP > AZ. Thus L and the 
circle meet at the single point A, which makes L a tangent.

That takes care of the "if" part.

Suppose now that L is not perpendicular to AZ, and drop a 
perpendicular from Z to L, meeting it at Q. Let the point R be on L 
such that RQ = AQ and RA = 2*AQ. Then show that point R lies on the 
circle. Thus L intersects the circle in two points, A and R, so is not 
tangent.

That takes care of the "only if" part.

You fill in the details and the reasons.

- Doctor Rob, The Math Forum
  http://mathforum.org/dr.math/   
    
Associated Topics:
High School Conic Sections/Circles
High School Geometry

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