S

Chapter 11 - Circle Theorems

Central and Inscribed Angles

  • Central Angle:
    • Vertex is at the center of the circle.
    • The measure of the central angle is twice the measure of the inscribed angle that subtends the same arc.
    • If the central angle is 120 degrees, it originates from the circle's center.
  • Inscribed Angle:
    • Vertex lies on the circle.
    • The measure of the inscribed angle is half the measure of the central angle that subtends the same arc.

Historical Context: Thales

  • Thales of Miletus (Phelous):
    • Lived approximately 2500 years ago.
    • Considered the father of ancient philosophy and science.
    • Teacher of Pythagoras.
    • Discovered that all inscribed angles subtending the same arc are equal.
    • Euclid referenced Thales's discoveries in his geometry book.
  • Pythagoras:
    • Lived near Thales and was about 50 years younger.
    • Learned mathematics from Thales.
    • Traveled to Egypt, Mesopotamia, and Ethiopia to expand his knowledge.

Properties of Inscribed Angles

  • Inscribed angles that intercept the same arc are equal.
  • Definition:
    • Inscribed angle: Vertex on the circle.
    • Central angle: Vertex at the center.
  • Formulas:
    • Central\ Angle = 2 \times Inscribed\ Angle
    • Inscribed\ Angle = \frac{1}{2} \times Central\ Angle

Chord Bisector Problem

  • Problem Setup:
    • A circle with center O.
    • A chord with a perpendicular bisector MN passing through the center.
    • OM = 3, radius (r) = 8 (since 5 + 3 = 8).
    • Goal: Find the length of MN.
  • Solution Steps:
    • Recognize that the radius is 8.
    • Form a right triangle using the radius and half the chord.
    • Use the Pythagorean theorem to find the missing length.
    • Let x be half the length of the chord. So, 5^2 + x^2 = 8^2. This implies, x = \sqrt{39}.
    • MN = 2\sqrt{39}.
    • Approximate 2\sqrt{39} \approx 12.4.
  • Key Insight:
    • The problem involves finding a hidden segment and applying the Pythagorean theorem.
  • Radius Identification:
    • Recognizing that the segments from the center to the circumference are radii is crucial.

Two Chord Product Theorem

  • Theorem Statement:
    • If two chords intersect inside a circle, the product of the segments of one chord equals the product of the segments of the other chord.
  • Formula:
    • If chords AB and CD intersect at point E, then AE \times EB = CE \times ED
  • Example:
    • Given segments with lengths 3, 5, 4, and x, where the chords intersect.
    • 3 \times 5 = 4 \times x
    • x = \frac{15}{4}
  • Application:
    • Use the formula to solve for unknown segment lengths when two chords intersect.

Secant-Secant Theorem

  • Theorem Statement:
    • If two secants intersect outside a circle, the product of the external segment and the total length of one secant equals the product of the external segment and the total length of the other secant.
  • Formula:
    • If secants intersect at point P outside the circle, then PA \times PB = PC \times PD, where A and B are points on one secant, and C and D are points on the other secant.
  • Example:
    • Given secant segments with lengths 4, x, 5, and 8.
    • 4 \times (4 + x) = 5 \times (5 + 8)
    • 16 + 4x = 65
    • 4x = 49
    • x = \frac{49}{4}
  • Mnemonic:
    • Short segment times the long segment is equal to the other short segment times the other long segment.
    • "Short, Long. Short, Long."

Secant-Tangent Theorem

  • Theorem Statement:
    • If a secant and a tangent intersect outside a circle, the product of the external segment of the secant and the total length of the secant equals the square of the length of the tangent segment.
  • Formula:
    • If a secant and a tangent intersect at point P outside the circle, then PA \times PB = PC^2, where A and B are points on the secant, and C is the point of tangency.
  • Example:
    • Given a secant with segments 4 and x, and a tangent of length 6.
    • 4 \times (4 + x) = 6^2
    • 4 \times (4 + x) = 36
    • 4 + x = 9
    • x = 5
  • Key Steps:
    • Recognize the segments and apply the formula.
    • Solve for the unknown length.

Tangent-Tangent Theorem

  • Theorem Statement:
    • If two tangents are drawn to a circle from an external point, then the segments from the external point to the points of tangency are congruent.
  • Example:
    • If two tangent segments have lengths 5 and x, then x = 5.
  • Properties:
    • The line from the center to the external point bisects the angle formed by the tangents.
    • The tangent line is perpendicular to the radius at the point of tangency (90 degrees).

Angles in a Circle

  • Inscribed Quadrilateral (Cyclic Quadrilateral):
    • Opposite angles are supplementary (add up to 180 degrees).
    • If one angle is 110 degrees, the opposite angle is 70 degrees.
    • Not covered in the exam but important to know.
  • Four Chord Angle:
    • Two chords intersecting inside the circle (Bow Tie Theorem).
    • Bow Tie Theorem Formula:
      • Angle = \frac{Arc1 + Arc2}{2}
    • If arc AB is 20 degrees and arc CD is 80 degrees, then the angle formed by the intersecting chords is (\frac{20 + 80}{2} = 50 \text{ degrees}). Plus sign inside intersection
  • Intersection Outside Circle:
    • Two chords intersecting outside the circle.
    • Formula:
      • Angle = \frac{Arc1 - Arc2}{2}
    • If arc CD is 70 degrees and arc AB is 10 degrees, then the angle formed by the intersecting chords is (\frac{70 - 10}{2} = 30 \text{ degrees}) Minus Sign outside circle intersection

Secant-Tangent Angle

  • Secant and Tangent Intersecting Outside Circle:
    • Consider only the arcs inside the "mouth" formed by the secant and tangent.
    • Formula:
      • Angle = \frac{BigArc - SmallArc}{2}
    • If arc BC is 160 degrees and arc AC is 60 degrees, then the angle formed by the secant and tangent is (\frac{160 - 60}{2} = 50 \text{ degrees}).