Answer :
The question is whether the arcs intersected by congruent chords [tex]\(\overline{AB}\)[/tex] and [tex]\(\overline{CD}\)[/tex] in two different circles are also congruent.
To answer this, let's analyze the situation step-by-step:
1. Consider the Two Circles: Let's denote the circles as Circle 1 and Circle 2, with centers [tex]\(O_1\)[/tex] and [tex]\(O_2\)[/tex] and radii [tex]\(r_1\)[/tex] and [tex]\(r_2\)[/tex], respectively. Assume [tex]\(r_1 \neq r_2\)[/tex].
2. Chords in Circles: Chord [tex]\(\overline{AB}\)[/tex] is in Circle 1, and chord [tex]\(\overline{CD}\)[/tex] is in Circle 2. Given [tex]\(\overline{AB} \cong \overline{CD}\)[/tex], the lengths of the chords [tex]\(\overline{AB}\)[/tex] and [tex]\(\overline{CD}\)[/tex] are equal.
3. Central Angles and Arcs: Draw the radii from the centers of the circles to the endpoints of the chords:
- In Circle 1, draw [tex]\(\overline{O_1A}\)[/tex] and [tex]\(\overline{O_1B}\)[/tex].
- In Circle 2, draw [tex]\(\overline{O_2C}\)[/tex] and [tex]\(\overline{O_2D}\)[/tex].
The chords [tex]\(\overline{AB}\)[/tex] and [tex]\(\overline{CD}\)[/tex] subtend central angles [tex]\(\angle AO_1B\)[/tex] and [tex]\(\angle CO_2D\)[/tex], respectively.
4. Triangles Formed: Observe the triangles formed by the radii and the chords:
- In Circle 1: [tex]\(\triangle O_1AB\)[/tex].
- In Circle 2: [tex]\(\triangle O_2CD\)[/tex].
These triangles are isosceles since [tex]\(\overline{O_1A} = \overline{O_1B} = r_1\)[/tex] and [tex]\(\overline{O_2C} = \overline{O_2D} = r_2\)[/tex].
5. Central Angles Comparison:
- Since [tex]\(\overline{AB}\)[/tex] and [tex]\(\overline{CD}\)[/tex] are congruent, the segments connecting the midpoints of [tex]\(\overline{AB}\)[/tex] and [tex]\(\overline{CD}\)[/tex] to the centers of their respective circles (perpendicular bisectors) are equal.
- However, the central angles [tex]\(\angle AO_1B\)[/tex] and [tex]\(\angle CO_2D\)[/tex] depend on the radii [tex]\(r_1\)[/tex] and [tex]\(r_2\)[/tex].
For isosceles triangles [tex]\(\triangle O_1AB\)[/tex] and [tex]\(\triangle O_2CD\)[/tex], if the radii [tex]\(r_1\)[/tex] and [tex]\(r_2\)[/tex] are different, the central angles [tex]\(\angle AO_1B\)[/tex] and [tex]\(\angle CO_2D\)[/tex] will not be the same.
6. Arc Length Determination: The measure of an arc in a circle is directly proportional to the central angle subtended by that arc. Therefore, if [tex]\(\angle AO_1B \neq \angle CO_2D\)[/tex], then the arcs intercepted by these chords will not have the same measure. Moreover, the arcs are different portions of their respective circles and thus inherit the difference from the circles' radii.
Conclusion: Since the central angles subtended by congruent chords in circles of different radii are not equal, the intercepted arcs are not congruent. Hence, the arcs intersected by chords [tex]\(\overline{AB}\)[/tex] and [tex]\(\overline{CD}\)[/tex] are not congruent.
To answer this, let's analyze the situation step-by-step:
1. Consider the Two Circles: Let's denote the circles as Circle 1 and Circle 2, with centers [tex]\(O_1\)[/tex] and [tex]\(O_2\)[/tex] and radii [tex]\(r_1\)[/tex] and [tex]\(r_2\)[/tex], respectively. Assume [tex]\(r_1 \neq r_2\)[/tex].
2. Chords in Circles: Chord [tex]\(\overline{AB}\)[/tex] is in Circle 1, and chord [tex]\(\overline{CD}\)[/tex] is in Circle 2. Given [tex]\(\overline{AB} \cong \overline{CD}\)[/tex], the lengths of the chords [tex]\(\overline{AB}\)[/tex] and [tex]\(\overline{CD}\)[/tex] are equal.
3. Central Angles and Arcs: Draw the radii from the centers of the circles to the endpoints of the chords:
- In Circle 1, draw [tex]\(\overline{O_1A}\)[/tex] and [tex]\(\overline{O_1B}\)[/tex].
- In Circle 2, draw [tex]\(\overline{O_2C}\)[/tex] and [tex]\(\overline{O_2D}\)[/tex].
The chords [tex]\(\overline{AB}\)[/tex] and [tex]\(\overline{CD}\)[/tex] subtend central angles [tex]\(\angle AO_1B\)[/tex] and [tex]\(\angle CO_2D\)[/tex], respectively.
4. Triangles Formed: Observe the triangles formed by the radii and the chords:
- In Circle 1: [tex]\(\triangle O_1AB\)[/tex].
- In Circle 2: [tex]\(\triangle O_2CD\)[/tex].
These triangles are isosceles since [tex]\(\overline{O_1A} = \overline{O_1B} = r_1\)[/tex] and [tex]\(\overline{O_2C} = \overline{O_2D} = r_2\)[/tex].
5. Central Angles Comparison:
- Since [tex]\(\overline{AB}\)[/tex] and [tex]\(\overline{CD}\)[/tex] are congruent, the segments connecting the midpoints of [tex]\(\overline{AB}\)[/tex] and [tex]\(\overline{CD}\)[/tex] to the centers of their respective circles (perpendicular bisectors) are equal.
- However, the central angles [tex]\(\angle AO_1B\)[/tex] and [tex]\(\angle CO_2D\)[/tex] depend on the radii [tex]\(r_1\)[/tex] and [tex]\(r_2\)[/tex].
For isosceles triangles [tex]\(\triangle O_1AB\)[/tex] and [tex]\(\triangle O_2CD\)[/tex], if the radii [tex]\(r_1\)[/tex] and [tex]\(r_2\)[/tex] are different, the central angles [tex]\(\angle AO_1B\)[/tex] and [tex]\(\angle CO_2D\)[/tex] will not be the same.
6. Arc Length Determination: The measure of an arc in a circle is directly proportional to the central angle subtended by that arc. Therefore, if [tex]\(\angle AO_1B \neq \angle CO_2D\)[/tex], then the arcs intercepted by these chords will not have the same measure. Moreover, the arcs are different portions of their respective circles and thus inherit the difference from the circles' radii.
Conclusion: Since the central angles subtended by congruent chords in circles of different radii are not equal, the intercepted arcs are not congruent. Hence, the arcs intersected by chords [tex]\(\overline{AB}\)[/tex] and [tex]\(\overline{CD}\)[/tex] are not congruent.
