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Chords and a Circle's Center - Concept
A chord is a line segment whose endpoints are on a circle. If a chord passes through the center of the circle, it is called a diameter. Two important facts about a circle chord are that (1) the perpendicular bisector of any chord passes through the center of a circle and (2) congruent chords are the same distance (equidistant) from the center of the circle.
Chords in the center of a circle have a special relationship but back up what's a chord? Let's refresh our memory. Well a chord is a line segment whose endpoints are on the circle. If I found the perpendicular bisector of this chord so if I took my compass and I swung arcs from both ends of that and I found the line that bisected this chord into two congruent pieces at a 90 degree angle, so let's say I do that in so this dotted line is my perpendicular bisector of that chord and no matter where I draw a chord on this circle if I find it's perpendicular bisector it will always pass through the center of the circle so that's the first key thing about a chord as relationship with the center of circle.
Let's talk about 2 congruent chords, so this is kind of a converse of what we just talked about. If I found the perpendicular bisector of these chords so if I measured the perpendicular distance from the chord to the center, so I'm going to draw a solid line here so this is the perpendicular distance because we said the shortest distance between two points is a line to perpendicular, if these chords are congruent, they will be the same distance away from the center of the circle so if I were to join two other chords and if I told you that these chords are congruent then their distance from the center of that circle measured along a perpendicular will be congruent. So using these two keys about chords and the relationship with the center will help us solve a lot of problems.