SOLUTION: How do you prove: in the same circle or in congruent circles, the shorter of two chords is the greater distance from the center of the circle?

Algebra ->  Circles -> SOLUTION: How do you prove: in the same circle or in congruent circles, the shorter of two chords is the greater distance from the center of the circle?      Log On


   

Question 371788: How do you prove: in the same circle or in congruent circles, the shorter of two chords is the greater distance from the center of the circle?
Answer by robertb(5830) About Me  (Show Source):
You can put this solution on YOUR website!
Consider a right triangle whose hypotenuse is the radius of the circle, and whose one leg is half of the chord as given. Then the 3rd leg is the perpendicular bisector from the center of the circle to the given chord.
Let the length of the given chord be c, and the radius, r. Then by the Pythagorean Theorem, the length of the 3rd leg is l+=+sqrt%28r%5E2+-+c%5E2%2F4%29. Noting that the radius r is fixed, then shortening c would have the effect of lengthening l, which is just the distance of the chord from the center of the circle.