Intersecting, Chords, Secants, and Tangents

Academic Vocabulary

  • Intersecting Chords

  • Intersecting Secants

  • Intersecting Secant and Tangent

Key Definitions

  • Chord: A line segment joining any two points on the circumference of the circle. The diameter is a special case of a chord.

  • Secant: A line that intersects a circle at two points, extending infinitely in both directions.

  • Tangent: A line that touches the circle at exactly one point, remaining outside the circle at all other points.

  • Intersection: The point where two lines, such as chords or secants, meet or cross each other, which can occur inside or outside the circle.

  • Intersecting Chords: When two chords intersect within a circle, they create segments that can be analyzed using the intersecting chords theorem, which states that the products of the lengths of the segments of each chord are equal to each other, allowing for the calculation of unknown lengths when certain segments are known.

Intersecting Chords Theorem

Part A

  • Theorem Description: When two chords intersect inside a circle, the measure of the angles formed is half the sum of the measures of the intercepted arcs.

  • Example Calculation:

    • Angles 1 & 3, 2 & 4 are vertical angles and thus congruent.

    • Formula: [ m\angle = \frac{m\text{(arc 1)} + m\text{(arc 2)}}{2} ]

  • Example Data: 120° and 84° lead to angle measures of 50° and 106°.

Application

  • Solve for the missing angle and arc measures using the theorem.

    • Example Result: [ m\angle UWV = 117° ]

Intersecting Chords Theorem (Product of Segments)

Part B

  • Theorem Description: When two chords intersect inside a circle, the products of their segments are equal.

  • Example Equation:

    • [ (6)(4) = (8)(x) \implies 24 = 8x \implies x = 3 ]

Application

  • Solve for missing segment lengths using this theorem.

    • Example Result: [ x = 12, TR = 27 ]

Intersecting Secants Theorem

Part A

  • Theorem Description: When two secants intersect inside a circle, the measure of the angles formed is half the difference of the measures of the intercepted arcs.

  • Example Calculation:

    • Formula: [ m\angle = \frac{m\text{(outside arc)} - m\text{(inside arc)}}{2} ]

  • Example Data: 116° and 50° yields angles calculated accordingly.

Application

  • Solve for the missing angle measures.

    • Example Result: [ m\angle QRS = 45°, m\angle HGF = 59° ]

Intersecting Secants Theorem (Product of Lengths)

Part B

  • Theorem Description: The product of the length of one secant segment and its external segment equals the product of the length of the other secant segment and its external segment.

  • Example Equation:

    • [ (6)(8+6) = (7)(x+7) \implies 84 = 7x + 49 \implies x = 5 ]

Application

  • Solve for missing segment lengths.

    • Example Results: [ x = 15, x = 9 ]

Intersecting Secant and Tangent Theorem

Part A

  • Theorem Description: When a secant and a tangent intersect inside a circle, the measure of the angles formed is half the difference of the measures of the intercepted arcs.

  • Example Data: 120° and 60°. The calculation is similar to that of intersecting arcs.

Application

  • Solve for angle measures using this theorem.

    • Example Result: [ m\angle MLK = 34° ]

Intersecting Secant and Tangent Theorem (Tangent Segment Length)

Part B

  • Theorem Description: The square of the length of the tangent segment is equal to the product of the length of the secant segment and the length of its external segment.

  • Example Equation:

    • [ QT^2 = (9)(9 + 16) \implies QT^2 = (9)(25) ]

Application

  • Solve for tangent lengths and radius.

    • Example Results: Diameter across segments equals values calculated from the equations.