Answer:
See Below.
Step-by-step explanation:
We are given two intersecting circles with centers M and N.
And we want to prove that: I) PQ ⊥ MN and that II) PR = RQ.
Since MP and MQ are radii of the same circle:
Likewise, since NP and NQ are radii of the same circle:
And by the Reflexive Property:
Therefore, by SSS Congruence:
By CPCTC:
And by the Reflexive Property:
And since they are the radii of the same circle:
Therefore, by SAS Congruence:
Therefore, by CPCTC:
Note that PQ is a chord in Circle M.
Therefore:
In a circle, a segment that passes through the center of the circle that is perpendicular to a chord also bisects the chord.
And conversely, a segment that passes through the center of a circle that bisects a chord in the circle is also perpendicular to the chord.
So:
