To analyze the given information about circle C and the relationships between the segments and chords, we can use some properties of circles and geometry.
- CG = CE indicates that the distances from point C to the chords FB and DA are equal.
- CG is perpendicular to FB indicates that the line segment CG is a radius that is perpendicular to the chord FB, meaning it bisects chord FB at point G.
- CE is perpendicular to DA indicates that the line segment CE is also a radius that is perpendicular to the chord DA, meaning it bisects chord DA at point E.
Given that CG and CE are equal and both segments are perpendicular to their respective chords, we can draw several conclusions based on properties of circles:
- Since both CG and CE bisect the chords FB and DA, respectively, we can say that segments FG is congruent to segment GB, and segment ED is congruent to segment EA.
- Since CG and CE are equal, we can analyze the triangles formed by these segments and the points on the circle.
However, among the provided choices:
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Segment FB is congruent to segment DA: This can be true if both chords are bisected by the same distance (which is typically true in this case), making FB and DA equal in length but only in specific cases, not necessarily a general conclusion.
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Segment DC is parallel to segment FA: There is no direct implication or information given that concludes segments DC are parallel to FA.
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Segment GC is parallel to segment CE: There is no relationship indicating that these segments are parallel.
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Segment FG is congruent to segment GC: Given that CG is equal to CE, we can conclude that FG = GC, as both segments are parts of the same perpendicular bisectors.
Since both segments FG and GC are parts of the triangles made by the perpendicular chords and that CG = CE, we can conclude:
Segment FG is congruent to segment GC.
This is the correct conclusion based on the relationships provided. Thus the best conclusion is the last option.
4 answers