The following figure shows 2 equidistant chords abdominal muscle and CD that a circle with center O. The ranges of the 2 chords room respectively OX and also OY:


We require to show that OX = OY. We have actually joined OA and also OC. Note that due to the fact that OX is perpendicular to AB, the must also bisect AB. Similarly, since OY is perpendicular come CD, it must likewise bisect CD. Thus, due to the fact that it is given that abdominal = CD, us have

< aise.5exhbox$scriptstyle 1$kern-.1em/kern-.15emlower.25exhbox$scriptstyle 2$ m left( AB ight) m = m AX m = m aise.5exhbox$scriptstyle 1$kern-.1em/kern-.15emlower.25exhbox$scriptstyle 2$ m left( CD ight) m = m CY>

Proof: to compare (Deltamathrm Omathrm Amathrm X) with(Deltamathrm Omathrm Cmathrm Y).

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1.OA = OC (radii that the same circle)

2.AX = CY (we simply proved this above)

3.(angle mOXA = angle mOYC = m 90^0)

By the RHS criterion, (Delta mOAX equiv Delta mOCY). Thus, OX = OY, which means that ab and CD room equidistant indigenous O.

The converse theorem also holds.

Converse: two chords the a circle which are equidistant from its facility must have actually the exact same length.

Proof: introduce to the previous figure again, we compare (DeltamathrmOAX) with (DeltamathrmOCY)

1.OA = OC (radii the the very same circle)

2.OX = OY (given)

3.(angle mOXA = angle mOYC = m 90^0)

By the RHS default again, the two triangles space congruent, and hence AX = CY. Since OX and also OY must additionally bisect abdominal and CD respectively (why?), us have

< mAB = m 2 mAX = m 2 mAY = m CD>

Thus, the 2 chords room of same length.

We now talk about a much more general result related come unequal chords.

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Theorem: For two unequal chords of a circle, the higher chord will certainly be nearer to the facility than the smaller sized chord.

Visually speaking, this need to be obvious. As your chord moves closer and closer come the center, it increases in length, together the following figure shows (the diameter is the largest possible chord in any circle):


The following figure shows two chords abdominal and CD that a one with center O, together that abdominal muscle > CD. OX and OY room perpendiculars come the 2 chords indigenous the center (this way that they will respectively bisect the two chords as well). We need to prove the OX YC (why?), this must median that