Chord Proofs

Date: 10/07/97 at 15:49:05
From: JENN BURKE
Subject: Chord Proofs

In any circle

a) a radius which is perpendicular to a chord also bisects the chord
   I tried adding in triangles but I don't know how to prove it.

b) a radius that bisects the chord is perpendicular to the chord
   I don't know how to prove that either.

c) chords that are equidistant from the center of the circle are 
   congruent
   Still can't prove it.

It would really be great if you could help me.  Thanks.

Date: 10/10/97 at 15:12:21
From: Doctor Rob
Subject: Re: Chord Proofs

In all cases, draw a diagram, such as the following:

                  .O
                 /|\
                / | \
               /  |  \
              /   |   \
             /    |    \
            /     |     \
           /      |      \
          /       |       \
         /        |C       \
      `-.---------.---------.-'
       A  ``--_________--''  B

               
   O is the center of the circle, AB is the chord.  
   OA = OB because they are radii of the circle.

a) OC is perpendicular to AB.  Use the Pythagorean Theorem in  
   triangles OCA and OCB.

b) OC bisects AB.  CA = CB. Use SSS to conclude that triangles OCA and
   OCB are congruent.  Then <ACO = <BCO.  But <ACO + <BCO = ... .

c) The length of OC is the distance from O to the chord AB. It is the
   perpendicular bisector of the chord AB. For a different chord DE,  
   with OF its perpendicular bisector, the distance from O to the 
   chord DE is the length of OF. You are given that the lengths of AB 
   and DE are the same. Then AC = AB/2 = DE/2 = DF.  Now use the 
   Pythagorean Theorem.

-Doctor Rob,  The Math Forum
 Check out our web site!  http://mathforum.org/dr.math/