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 1)Construct two intersecting chords in a circle. 3)Draw PS and RQ. 2)Name the chords PQ and RS intersecting at T. S P Q R T

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

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

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

Example 2 Intersection of Two Chords Find x x

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

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

SEGMENTS INTERSECTING OUTSIDE A CIRCLE Theorem E B C A D 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.

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

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

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. Z W X Y

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

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