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Special Segments in a Circle

Special Segments in a Circle. Find measures of segments that intersect in the interior of a circle. Find measures of segments that intersect in the exterior of a circle. A Tibetan Mandala exhibiting a six-pointed star. SEGMENTS INTERSECTING INSIDE A CIRCLE. R.

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Special Segments in a Circle

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  1. Special Segments in a Circle • Find measures of segments that intersect in the interior of • a circle. • Find measures of segments that intersect in the exterior of • a circle. A Tibetan Mandala exhibiting a six-pointed star.

  2. SEGMENTS INTERSECTING INSIDE A CIRCLE R Construct two intersecting chords in a circle. P Name the chords PQ and RS intersecting at T. T S Q Draw PS and RQ.

  3. SEGMENTS INTERSECTING INSIDE A CIRCLE R P Analyze: PTS  RTQ Vertical Angles T S Q P  R Angles intercept the same arc By angle-angle similarity, or PT ∙ TQ = RT ∙ ST

  4. Theorem R If two chords intersect in a circle, then the products of the measures of the segments of the chords are equal. P T S Q or PT ∙ TQ = RT ∙ ST

  5. Example 1 Intersection of Two Chords D Find x 4 A x E 6 3 C B

  6. Example 2 Intersection of Two Chords Find x x 9 12 8

  7. Example 3 Solve Problems What is the radius of the circle containing the arc if the arc is not a semicircle? 12 24 24

  8. Example 3 continued Solve Problems What is the radius of the circle containing the arc if the arc is not a semicircle? 12 24 24 Solution: 24 x 24 = 12x 576 = 12x 48 = x x Diameter = 48 + 12 = 60 Radius = 60/2 = 30

  9. C B A E D SEGMENTS INTERSECTING OUTSIDE A CIRCLE Theorem If two secant segments are drawn to a circle from an exterior point, then the product of the measures of one secant segment and its external secant segment is equal to the product of the measures of the other secant segment and its external secant segment.

  10. Example 4 Intersection of Two Secants Find RS if PQ = 12, QR = 2, and TS = 3. 12 2 P R Let RS = x Q x S 3 T Disregard the negative value

  11. Example 5 Intersection of Two Secants Find x if EF = 10, EH = 8, and FG = 24. 8 E x 10 H F I 24 G

  12. X W Z Y Theorem If a tangent segment and a secant segment are drawn to a circle from an exterior point, then the square of the measure of the tangent segment is equal to the product of the measures of the secant segment and its external secant segment.

  13. A B C D Example 6 Intersection of a Secant and a Tangent Find x. 4 x x + 2 The expression is not factorable. Use the quadratic formula. or Disregard the negative solution

  14. Example 7 Intersection of a Secant and a Tangent Find x. x + 2 x x + 4 Disregard the negative value

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