NC.M3.G-C.2
Circle Theorems
Central and inscribed angles, tangents and chords, and the angles formed when secants and tangents meet — inside or outside the circle.
Source: NC Math 3 · Geometry Study Guide (PDF prototype), Part 2, pages 2-3
Central and inscribed angles
Central angle (vertex at the center) = its arc — ∠AOB = arc AB
Inscribed angle (vertex on the circle) = half its arc — ∠ACB = ½ · arc AB
Consequences worth memorizing:
- An inscribed angle that cuts off a semicircle (a diameter) is 90°.
- Inscribed angles that cut off the same arc are equal.
- A quadrilateral inscribed in a circle has opposite angles that add to 180°.
- The whole circle is 360°.
Tangents and chords
Tangent ⊥ radius
A tangent meets the radius at a right angle at the point of tangency. This often sets up a Pythagorean Theorem problem.
Two tangents from the same outside point are equal: PA = PB.
Example
r = 5 and OP = 13. Find PA.
5² + PA² = 13² → PA² = 144 → PA = 12
5² + PA² = 13² → PA² = 144 → PA = 12
Radius ⊥ chord bisects the chord
If a radius (or diameter) is perpendicular to a chord, it cuts the chord into two equal halves: AM = MB.
Triangle OMB is a right triangle with hypotenuse r.
Example
r = 10 and OM = 6. Find chord AB.
MB² + 6² = 10² → MB = 8 → AB = 16
MB² + 6² = 10² → MB = 8 → AB = 16
Segment products: "part × part" and "outside × whole"
Two chords that cross
the two pieces of one chord multiply to the same thing as the two pieces of the other — a · b = c · d
Example
a = 4, b = 6, c = 3. Find d.
4 · 6 = 3d → d = 8
4 · 6 = 3d → d = 8
Secant or tangent from an outside point
Two secants — outside · whole = outside · whole
Tangent and secant — t² = outside · whole
Example
outside = 4, inside = 5. Find t.
whole = 4 + 5 = 9 → t² = 4 · 9 = 36 → t = 6
whole = 4 + 5 = 9 → t² = 4 · 9 = 36 → t = 6
Trap: For the outside-point rules, use the whole secant (outside part + inside part), not just the inside part.
Angles formed by chords, secants, and tangents
Vertex INSIDE the circle (two chords cross)
half the sum of the arc the angle cuts off and the arc its vertical angle cuts off — angle = ½ (x + y)
Example
arcs of 70° and 110°
½(70 + 110) = 90°
½(70 + 110) = 90°
Vertex OUTSIDE the circle (secants or tangents)
half the difference; works for any mix of secants and tangents — angle = ½ (far arc − near arc)
Example
far arc 150°, near arc 50°
½(150 − 50) = 50°
½(150 − 50) = 50°
Memory hook: The farther the vertex is from the center, the smaller the angle. Center: the whole arc. Inside: half the sum. Outside: half the difference.
Practice
1. An inscribed angle cuts off an arc of 140°. What is the angle?
2. AB is a diameter and C is on the circle. If ∠CAB = 35°, find ∠CBA.
3. PT is tangent to circle O at T. The radius is 8 and PT = 15. Find PO.
4. Two tangents from point P touch the circle at A and B. PA = 3x + 2 and PB = 5x − 6. Find x and PA.
5. Chords AB and CD cross at E. AE = 5, EB = 8, CE = 4. Find ED.
6. A chord of length 24 is in a circle of radius 13. How far is the chord from the center?
7. Two chords cross inside a circle. The arcs cut off by the angle and its vertical angle are 84° and 36°. Find the angle.
8. Two secants meet at a point outside a circle. The far arc is 130° and the near arc is 40°. Find the angle.