Question

when two chords intersect, four line segments are created. what relationship exists between the lengths of those line segments?

Answer 1

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Explanation:

The product of the lengths of the segments of one chord is equal to the product of the lengths of the segments of the other chord. (The lengths are measured from the point of intersection of the chords to the points of intersection of the chord with the circle.)

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Additional comment

This relationship can be generalized to include the situation where the point of intersection of the lines is outside the circle. In that geometry, the lines are called secants, and the segment measures of interest are the measures from their point of intersection to the near and far intersection points with the circle. Again, the product of the segment lengths is the same for each secant.

This can be further generalized to the situation where the two points of intersection of one of the secants are the same point--the line is a tangent. In that case, the segment lengths are both the same, so their product is the square of the length of the tangent from the circle to the point of intersection with the secant.

So, one obscure relationship can be generalized to cover the relationships between segment lengths in three different geometries. I find it easier to remember that way.