10'5 Segment Lengths in Circles

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10.5 Segment Lengths in Circles

• 21.0 Students prove and solve problems regarding
  relationships among chords, secants, tangents,
  inscribed angles, and inscribed and circumscribed
  polygons of circles.

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Warmup

• What is the formula for when for the angle or
  its vertical angle formed by a chord?
• It is the average of the sum of intercepted arcs
  made by the two chords.
• What is the formula for when 2 lines intersect
  outside a circle, but are tangent, secant, or
  both tangent or both secant to a circle?
• The angle created is the average of the
  difference of the two intercepted arcs created by
  the two intersecting lines.

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Review Problems
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• AB, CD chords intersect at E.
• Draw AC and BD
• ltACE is congruent to ltDBE.
• ltCEA is congruent to ltBED.
• ?CEA?BED
• CE x ED EA x EB

• Given
• 2 points determine a line.
• If two points intercept the same arc they are
  congruent.
• Vertical angles are congruent.
• AA
• Corresponding sides of similar triangles are
  proportional.
• Cross product property

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Now you practice!
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Drawing Activity

• Draw a circle. Then draw an exterior point label
  it point E.
• Now draw a secant from that point to the circle.
  Label the 1st and 2nd intersection with the
  circle A and B respectively.
• Draw another secant from the same exterior point
  to the circle. Label the 1st and 2nd intersection
  with the circle C and D respectively.
• Measure the segment EA and AB then take the
  product of these two measurements.
• Measure the segment EC and ED, then take the
  product of these two measurements.
• What is their relationship?
• Conjecture?

B
A
E
D
C
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Theorem 10.16

• If two secant segments share the same endpoint
  outside a circle, then the product of the length
  and 1 secant segment and the length of its
  external segment equals the product of the length
  of the other secant segment and the length of its
  external segment.

B
A
E
D
C
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Practice problem
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Theorem 10.17

• Draw a circle. Then draw an exterior point label
  it point E.
• Now draw a tangent from that point to the circle.
  Label the tangent point A.
• Draw a secant from the same exterior point E to
  the circle. Label the 1st and 2nd intersection
  with the circle C and D respectively.
• Measure the segment EA the square it.
• Measure the segment EC and ED, then take the
  product of these two measurements.
• What is their relationship?
• Conjecture?

A
E
D
C
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Theorem 10.17

• In a secant segment and a tangent segment share
  an endpoint outside a circle, then the product of
  the length of the secant segment and the length
  of its external segment equals the square of the
  length of the tangent segment.

A
E
D
C
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Practice problem.
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More practice!
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Homework 10.5

• Page 632 (2-8, 10-17all, 19-23, 25)