Tangents to CirclesDate: 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/ |
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