Special Segments in a Circle Find measures of

  • Slides: 14
Download presentation
Special Segments in a Circle • Find measures of segments that intersect in the

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. 2)

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

SEGMENTS INTERSECTING INSIDE A CIRCLE R P Analyze: PTS RTQ Vertical Angles T S

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

Theorem If two chords intersect in a circle, then the products of the measures

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

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

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

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

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

Example 3 What is the radius of the circle containing the arc if the

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

Example 3 continued What is the radius of the circle containing the arc if

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

SEGMENTS INTERSECTING OUTSIDE A CIRCLE Theorem B C A E D If two secant

SEGMENTS INTERSECTING OUTSIDE A CIRCLE Theorem B C A E 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 4 Intersection of Two Secants Find RS if PQ = 12, QR =

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

Example 5 Intersection of Two Secants Find x if EF = 10, EH =

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

Theorem X W Z Y If a tangent segment and a secant segment are

Theorem X W Z Y 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.

Example 6 Intersection of a Secant and a Tangent Find x. A 4 B

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

Example 7 Find x. Intersection of a Secant and a Tangent x+2 x x+4

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