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Tangent, Secant, and Chord Angle Theorems

A free Geometry lesson from the “Measurement, Circles, and 3D Solids” unit, with a worked example and practice problems including step-by-step solutions.

Inscribed angles equal half the intercepted arc. Two chords meeting INSIDE a circle form an angle equal to half the SUM of the two intercepted arcs. Two secants (or a tangent + secant, or two tangents) meeting OUTSIDE a circle form an angle equal to half the DIFFERENCE of the two intercepted arcs. The power of a point gives equal products: chord-chord AE * EB = CE * ED; tangent-secant tangent^2 = whole-secant * near-piece.

What you'll learn

Why it matters: Optics (reflection off curved surfaces), satellite footprint calculations, and structural engineering of arches all use circle-angle theorems to relate angles, arcs, and segment lengths.

Worked example

Problem. Two chords AB and CD intersect inside a circle at E. AE = 2, EB = 6, CE = 3. Find ED.

  1. Intersecting chords: AE * EB = CE * ED.
  2. 2 * 6 = 3 * ED, so 12 = 3 * ED.
  3. ED = 4.

Answer: 4

Practice problems

1. An inscribed angle intercepts an arc of 100 degrees. Find the angle in degrees.

Show solution
  1. Warm-up: First identify exactly what the question is asking: An inscribed angle intercepts an arc of 100 degrees. Find the angle in degrees.
  2. For intercepts, remember that an x-intercept has y = 0 and a y-intercept has x = 0.
  3. Inscribed angle = arc / 2.
  4. 100 / 2 = 50.
  5. Check the result by substituting or estimating: the response should match 50 and make sense in the original problem.

Answer: 50

2. A central angle intercepts an arc of 80 degrees. Find the central angle.

Show solution
  1. Warm-up: First identify exactly what the question is asking: A central angle intercepts an arc of 80 degrees. Find the central angle.
  2. For intercepts, remember that an x-intercept has y = 0 and a y-intercept has x = 0.
  3. A central angle equals its intercepted arc.
  4. Check the result by substituting or estimating: the response should match 80 and make sense in the original problem.

Answer: 80

3. A tangent-chord angle intercepts an arc of 80 degrees. Find the angle.

Show solution
  1. Warm-up: First identify exactly what the question is asking: A tangent-chord angle intercepts an arc of 80 degrees. Find the angle.
  2. For intercepts, remember that an x-intercept has y = 0 and a y-intercept has x = 0.
  3. Tangent-chord angle = arc / 2.
  4. 80 / 2 = 40.
  5. Check the result by substituting or estimating: the response should match 40 and make sense in the original problem.

Answer: 40

4. Two chords inside a circle intercept arcs of 60 and 100 degrees. Find the angle they form.

Show solution
  1. Core Practice: First identify exactly what the question is asking: Two chords inside a circle intercept arcs of 60 and 100 degrees. Find the angle they form.
  2. For intercepts, remember that an x-intercept has y = 0 and a y-intercept has x = 0.
  3. Inside the circle: angle = (60 + 100) / 2.
  4. = 160 / 2 = 80.
  5. Check the result by substituting or estimating: the response should match 80 and make sense in the original problem.

Answer: 80

5. Two secants from an external point intercept arcs of 120 and 40. Find the angle.

Show solution
  1. Core Practice: First identify exactly what the question is asking: Two secants from an external point intercept arcs of 120 and 40. Find the angle.
  2. For intercepts, remember that an x-intercept has y = 0 and a y-intercept has x = 0.
  3. Outside the circle: angle = (120 - 40) / 2.
  4. = 80 / 2 = 40.
  5. Check the result by substituting or estimating: the response should match 40 and make sense in the original problem.

Answer: 40

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