Tenth Grade · Math · week 33

Circles: segment relationships and the equation of a circle

Apply the chord-chord, secant-secant, and tangent-secant segment relationships, and write or interpret the equation of a circle.

Three segment relationships handle chords, secants, and tangents that meet inside or outside a circle. If two chords intersect INSIDE a circle, the two products of their segments are equal: (piece 1)(piece 2) = (piece 3)(piece 4). If two secants meet OUTSIDE a circle, multiply each secant's external segment by its own whole length, and those two products are equal. If a tangent and a secant meet outside a circle, the tangent segment squared equals the secant's external segment times its whole length -- the tangent behaves like a secant whose "external" and "whole" pieces collapse into one length.

A circle also has an algebraic identity: the equation of a circle with center (h, k) and radius r is (x - h)^2 + (y - k)^2 = r^2 -- literally the distance formula, squared, applied to every point (x, y) sitting exactly r units from the center. Given an equation in this form, read off the center and radius directly; given a center and a radius, write the equation directly.

📋 For the teacher
Teach it (10–15 min):
  • Draw all three cases (chord-chord, secant-secant, tangent-secant) on one page side by side, since where the pieces meet (inside versus outside) is what determines which formula to use.
  • For the tangent-secant case, point out the tangent segment gets SQUARED because it plays the role of BOTH the external piece and the whole length at once.
  • For the circle equation, drill reading off h, k, and r from an equation FIRST (including the sign flip: (x-3) means h=3, not h=-3) before writing equations from a description.
  • Connect the circle equation back explicitly to the distance formula from earlier this year -- this is a familiar idea applied to every point on a circle at once.
Talk about it:
  • Why does the tangent-secant relationship square the tangent length instead of multiplying two different segments?
  • Why does the sign inside the circle equation's parentheses flip from the center's actual coordinates?
  • How are the three segment relationships similar to each other?

Mastery looks like: They correctly apply the chord-chord, secant-secant, and tangent-secant segment relationships to solve for unknown lengths, and write or interpret the standard equation of a circle.

Common stumbles: Multiplying the wrong pairs of segments (not matching each secant's own external-times-whole); forgetting to flip the sign when reading the center from an equation.

See it visually

Interactive, spoken-aloud tools that show this idea in motion.

Unit CircleAngles, radians, and sine & cosine come alive.try →
Homework for this lessonType it, write it with a stylus, or print it for pencil & paper.open →
Records attendance + 15 min of Math for this learner.
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