DWG NO. 12.4 — Lesson 4 of 5

Chords, Secants, and Segment Relationships

Unit 12: Circles · ~15–30 min

Objective

Find unknown angle and segment measures formed by intersecting chords, secants, and tangents.

Angles formed inside and outside a circle

When two chords, secants, or tangents cross, the angle they form is tied to the arcs they cut off — whether they meet inside the circle or outside it changes whether you add or subtract those arcs.

Two chords intersecting inside the circle: the angle formed equals half the sum of the two intercepted arcs.
Two secants, two tangents, or a secant and tangent meeting outside the circle: the angle formed equals half the difference of the two intercepted arcs (far arc minus near arc).

Segment length relationships

Intersecting Chords Theorem: when two chords intersect inside a circle, the products of their segments are equal — if chord one splits into pieces a and b, and chord two splits into pieces c and d, then a · b = c · d.
Two Secants Theorem: from an external point, (whole secant)(external part) = (whole secant)(external part) for each secant.
Secant–Tangent Theorem: from an external point, (tangent)² = (whole secant)(external part).

Chords AB and CD intersect at P; PA · PB = PC · PD, and ∠APC = ½(arc AC + arc BD).

Worked Example 1 · Intersecting chords

ProblemChords AB and CD intersect at P inside a circle. PA = 8, PB = 6, PC = 4. Find PD.

1.By the Intersecting Chords Theorem: PA · PB = PC · PD.

2.Substitute known values: 8(6) = 4(PD), so 48 = 4(PD).

3.Solve: PD = 12.

PD = 12

Worked Example 2 · Angle from two secants

ProblemTwo secants meet at external point E. The far arc they intercept measures 118° and the near arc measures 34°. Find the angle at E.

1.For two secants meeting outside the circle, the angle is half the difference of the intercepted arcs: m∠E = ½(118° − 34°).

2.Simplify: m∠E = ½(84°).

m∠E = 42°

Guided practice

1.Two chords intersect inside a circle. One chord splits into segments 5 and 9; the other splits into segments 3 and x. Find x.
Show answer
5(9) = 3(x) → 45 = 3x → x = 15.
2.Two chords intersect inside a circle, cutting off arcs of 76° and 52° on opposite sides of the vertex. Find the measure of the angle formed.
Show answer
Angle = ½(sum of intercepted arcs) = ½(76 + 52) = ½(128) = 64°.
3.A tangent and a secant meet at external point F. The tangent segment measures 12, the secant's external part measures 8. Find the whole length of the secant.
Show answer
tangent² = (whole secant)(external part) → 12² = (whole)(8) → 144 = 8(whole) → whole = 18.
4.Two secants from an external point intercept a far arc of 150° and a near arc of 40°. Find the angle formed at the external point.
Show answer
Angle = ½(far arc − near arc) = ½(150 − 40) = ½(110) = 55°.
5.Explain in your own words why the chords formula uses a sum of arcs while the external-point formula uses a difference of arcs.
Show answer
When the vertex is inside the circle, both arcs lie on either side of the angle and both "contribute" to closing the angle, so their measures add. When the vertex is outside the circle, one arc is farther away and one is closer — the near arc effectively cancels part of the far arc's contribution, so the formula subtracts them.

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