Answer :
To determine whether the statement "Two arcs of a circle are congruent if and only if their associated chords are perpendicular" is true or false, we need to analyze the relationship between congruent arcs and their associated chords.
Step-by-step analysis:
1. Understanding Congruent Arcs:
- Two arcs are said to be congruent if they have the same measure. This implies that both arcs subtend the same central angle in the circle.
2. Congruent Arcs and Their Chords:
- If two arcs are congruent, they subtend equal angles at the center of the circle.
- The chords corresponding to these arcs would also be congruent, meaning they have the same length.
3. Perpendicular Chords:
- For two chords to be perpendicular, they must intersect at a right angle (90 degrees).
- However, having perpendicular chords does not necessarily mean that the arcs subtended by these chords are congruent. Perpendicularity is an independent condition related to the angle between the chords and does not imply equal arc measures.
4. Analysis of the Statement:
- The statement claims that two arcs are congruent if and only if their associated chords are perpendicular. This implies a bidirectional condition:
1. If the arcs are congruent, their chords need to be perpendicular.
2. If the chords are perpendicular, the arcs need to be congruent.
- While the first part is false because congruent arcs imply equal chords but not necessarily perpendicularity, the second part is also false since perpendicular chords do not guarantee congruent arcs.
Conclusion:
Based on the above reasoning, the correct answer to the question is:
B. False
Step-by-step analysis:
1. Understanding Congruent Arcs:
- Two arcs are said to be congruent if they have the same measure. This implies that both arcs subtend the same central angle in the circle.
2. Congruent Arcs and Their Chords:
- If two arcs are congruent, they subtend equal angles at the center of the circle.
- The chords corresponding to these arcs would also be congruent, meaning they have the same length.
3. Perpendicular Chords:
- For two chords to be perpendicular, they must intersect at a right angle (90 degrees).
- However, having perpendicular chords does not necessarily mean that the arcs subtended by these chords are congruent. Perpendicularity is an independent condition related to the angle between the chords and does not imply equal arc measures.
4. Analysis of the Statement:
- The statement claims that two arcs are congruent if and only if their associated chords are perpendicular. This implies a bidirectional condition:
1. If the arcs are congruent, their chords need to be perpendicular.
2. If the chords are perpendicular, the arcs need to be congruent.
- While the first part is false because congruent arcs imply equal chords but not necessarily perpendicularity, the second part is also false since perpendicular chords do not guarantee congruent arcs.
Conclusion:
Based on the above reasoning, the correct answer to the question is:
B. False
