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Circles and Lengths of Segments

Circles and Lengths of Segments. December 1, 2009. A bit of review: Inscribed angles. Theorem: The measure of an inscribed angle is equal to half the measure of its intercepted arc. Inscribed quadrilaterals.

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Circles and Lengths of Segments

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  1. Circles and Lengths of Segments December 1, 2009

  2. A bit of review: Inscribed angles • Theorem: The measure of an inscribed angle is equal to half the measure of its intercepted arc.

  3. Inscribed quadrilaterals • If a quadrilateral is inscribed in a circle, then its opposite angles are supplementary.

  4. Angles formed by two chords • Theorem: The measure of an angle formed by two chords that intersect inside a circle is equal to half the sum of the measures of the intercepted arcs. • Angle 1 = ½ * (CA+BD) • Angle 1 = angle 3 + angle 2

  5. Angles formed by secants, tangents, or both • Theorem: The measure of an angle formed by two secants, two tangents, or a secant and a tangent drawn from a point outside a circle is equal to half the difference of the measures of the intercepted arcs. • Angles 1, 2, and 3 each = ½ * (x-y)

  6. Something new: segments of chord • In the figure below, chords AB and CD intersect inside circle O. AM and MB are called the segments of chord. • The phrase "segment of a chord" will refer to the length of a segment as well as the segment itself (in the same way that we use the terms "radius" and "diameter").

  7. The relationship between segments of chord • Theorem: When two chords intersect inside a circle, the product of the segments of one chord equals the product of the segments of the other chord. • r * s = t * u

  8. Secant Segments • Theorem: When two secant segments are drawn to a circle from an external point, the product of one secant segment and its external segment equals the product of the other secant segment and its external segment. • r * s = t * u

  9. Expanding to secant segments and tangent segments • r * s = t * u

  10. Secant and tangent • Theorem: When a secant segment and a tangent segment are drawn to a circle from an external point, the product of the secant segment and its external segment is equal to the square of the tangent segment. (r*s= t2)

  11. Try this one • Find x.

  12. And this one.. • Find x.

  13. Another one.. • Find x.

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