In a circle, a pair of chords that subtend congruent (equal) central angles must be:
congruent (equal in length)
perpendicular to each other
parallel to each other
of different lengths
Hint 1: Draw radii from the center to the endpoints of both chords to form two triangles.
Hint 2: Apply the Side-Angle-Side (SAS) congruence theorem to these two triangles.
Hint 3: Since the triangles are congruent, their third sides (the chords) must be equal in length. This is at option index 0.
Step 1 (Construct triangles): Let the circle have center . Let chord 1 be and chord 2 be . Both chords subtend congruent central angles, so:
Step 2 (Apply Side-Angle-Side Congruence): Consider and :
(radii of the circle)
(radii of the circle)
(given congruent central angles)
By the Side-Angle-Side (SAS) congruence criterion:
Step 3 (Corresponding parts of congruent triangles): Since the triangles are congruent, their corresponding side lengths must be equal:
This proves that the two chords are congruent (equal in length).
Step 4 (Conclusion): The chords must be congruent (equal in length), corresponding to option index 0.
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