Segment-Length Relationships in Circles
When lines cross inside or outside a circle, the pieces they cut off are not random — their products are equal. There are three versions of the same idea, one for two chords, one for two secants, and one for a secant meeting a tangent.
Each rule turns a picture into an equation you can solve for a missing length. The hard part is reading which pieces to multiply, so this comes down to labeling the segments carefully.
Two chords that intersect inside
If chords and cross at point inside the circle, then . Each side of the equation is one chord split into its two pieces, multiplied together.
The two products are always equal, no matter where inside the circle the chords cross. Set them equal and solve for whichever piece is unknown.
Secants and tangents from an outside point
From a point outside the circle, two secants give (whole secant one)(external piece one) = (whole secant two)(external piece two). The key is that you multiply the entire secant by only its outside piece. In the figure, is the tangent, while the secant runs from through the near point to the far point .
If one line is a tangent instead, its length is used twice: . A tangent has no inside piece, so it plays the role of both factors.
Worked examples
Example 1: two intersecting chords
Chords cross so that one is split into and , the other into and . Find .
Answer:
Example 2: a tangent and a secant
A tangent of length and a secant with external piece share an outside point. Find the secant's whole length .
Answer:
Example 3: two secants from the same outside point
One secant has external piece and whole length . A second secant from the same point has external piece . Find its whole length .
Answer:
Try one yourself
Common questions
How do I tell which rule to use?
Look at where the lines meet. Inside the circle means two chords (). Outside means secants or a tangent, and you use whole-times-external pieces.
Why is the tangent used twice?
A tangent touches the circle at exactly one point, so its inside piece is zero length. The rule is the secant rule with both pieces of the tangent equal to its full length.
Does external piece mean the part outside the circle?
Yes — the segment from the outside point to where the secant first meets the circle. The whole secant runs from that outside point all the way to the far side.
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