If one chord of a circle is a perpendicular bisector of another chord, then the first chord is a diameter.
Now we will explain how you can connect these two theorems. Let's consider the following diagram.
We can see that QS is a perpendicular bisector of TR. Therefore, QS is a diameter of the circle by the Perpendicular Chord Bisector Converse.
Knowing that QS is a diameter and the fact that it is perpendicular to TR, we have that QS bisects TR — which we already knew — and that QS bisects the arc RT, RQ≅QT by the Perpendicular Chord Bisector Theorem. In conclusion, we have proven the following.
If one chord of a circle is a perpendicular bisector of another chord, then the first chord is a diameter and it bisects the arc of the second chord.
Describing and Correcting the Error
Finally, let's consider the given diagram.
The exercise states the following.
BecauseACbisectsDB, BC≅CD.
It seems like the Perpendicular Chord Bisector Converse and then the Perpendicular Chord Bisector Theorem were used in the way we described. However, the assumptions of these theorems were not met. The chord AC bisects DB, but it is notperpendicular to DB. This means we cannot conclude that AC bisects BD.
ACbisectsDB ⇒ BC≅CD
From what we know, we cannot assume that the arcs BC and CD are congruent.
