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Equal Chords are Equidistant from the Centre
Equal Chords are Equidistant from the Centre
This property states that if 2 chords have the same length, they are equidistant from the centre.
This property states that if 2 chords have the same length, they are equidistant from the centre.
Conversely, if 2 chords are equidistant from the centre, the chords must have the same length.
Conversely, if 2 chords are equidistant from the centre, the chords must have the same length.
This is found in Euclid Elements: Book III Proposition XIV.
This is found in Euclid Elements: Book III Proposition XIV.
Proof
Proof
This property (and its converse) and both be proven by using congruent triangles.
This property (and its converse) and both be proven by using congruent triangles.
The green and yellow triangles shown in the diagram are congruent by the RHS test. The details are left to the reader.
The green and yellow triangles shown in the diagram are congruent by the RHS test. The details are left to the reader.
Explore!
Explore!
Move points A and B about to adjust the length and location of chord AB.
Move points A and B about to adjust the length and location of chord AB.
Check on 'Show Distance' to see the red dashed line that represents the distance of the chord to the centre.
Check on 'Show Distance' to see the red dashed line that represents the distance of the chord to the centre.
Check on '2nd Chord' to move about a 2nd chord to see how two chords that are equidistant from the centre have the same length (and vice-versa).
Check on '2nd Chord' to move about a 2nd chord to see how two chords that are equidistant from the centre have the same length (and vice-versa).
Next: The radius is the perpendicular bisector of a chord.
Next: The radius is the perpendicular bisector of a chord.
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