Circles and Lengths of Segments

December 1, 2009Circles and Lengths of Segments1A bit of review: Inscribed anglesTheorem: The measure of an inscribed angle is equal to half the measure of its intercepted arc.

2Inscribed quadrilateralsIf a quadrilateral is inscribed in a circle, then its opposite angles are supplementary.

3Angles formed by two chordsTheorem: 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 = * (CA+BD)Angle 1 = angle 3 + angle 2

4Angles formed by secants, tangents, or bothTheorem: 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 = * (x-y)

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16Something new: segments of chordIn 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").

17The relationship between segments of chordTheorem: 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

18Secant SegmentsTheorem: 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

19Expanding to secant segments and tangent segmentsr * s = t * u

20Secant and tangentTheorem: 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)

21Try this oneFind x.

22And this one..Find x.

23Another one..Find x.

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