10.5 Segment Lengths in Circles

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10.5 Segment Lengths in Circles

• Geometry
• Mrs. Spitz
• Spring 2005

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Objectives/Assignment

• Find the lengths of segments of chords.
• Find the lengths of segments of tangents and
  secants.
• Assignment pp. 632-633 1-30 all.
• Practice Quiz pg. 635 1-7 all.

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Finding the Lengths of Chords

• When two chords intersect in the interior of a
  circle, each chord is divided into two segments
  which are called segments of a chord. The
  following theorem gives a relationship between
  the lengths of the four segments that are formed.

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Theorem 10.15

• If two chords intersect in the interior of a
  circle, then the product of the lengths of the
  segments of one chord is equal to the product of
  the lengths of the segments of the other chord.

EA EB EC ED
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Proving Theorem 10.15

• You can use similar triangles to prove Theorem
  10.15.
• Given , are chords that intersect at
  E.
• Prove EA EB EC ED

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Proving Theorem 10.15

• Paragraph proof Draw and .
  Because ?C and ?B intercept the same arc, ?C ?
  ?B. Likewise, ?A ? ?D. By the AA Similarity
  Postulate, ?AEC ? ?DEB. So the lengths of
  corresponding sides are proportional.

Lengths of sides are proportional.
EA EB EC ED
Cross Product Property
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Ex. 1 Finding Segment Lengths

• Chords ST and PQ intersect inside the circle.
  Find the value of x.

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Using Segments of Tangents and Secants

• In the figure shown, PS is called a tangent
  segment because it is tangent to the circle at an
  end point. Similarly, PR is a secant segment and
  PQ is the external segment of PR.

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Theorem 10.16

• If two secant segments share the same endpoint
  outside a circle, then the product of the length
  of one 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.

EA EB EC ED
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Theorem 10.17

• If 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 equal the square of the
  length of the tangent segment.

(EA)2 EC ED
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Ex. 2 Finding Segment Lengths

• Find the value of x.

RP RQ RS RT
Use Theorem 10.16
Substitute values.
9(11 9)10(x 10)
Simplify.
180 10x 100
Subtract 100 from each side.
80 10x
Divide each side by 10.
8 x
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Note

• In Lesson 10.1, you learned how to use the
  Pythagorean Theorem to estimate the radius of a
  grain silo. Example 3 shows you another way to
  estimate the radius of a circular object.

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Ex. 3 Estimating the radius of a circle

• Aquarium Tank. You are standing at point C,
  about 8 feet from a circular aquarium tank. The
  distance from you to a point of tangency is about
  20 feet. Estimate the radius of the tank.

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Solution
(CB)2 CE CD
Use Theorem 10.17
(20)2 ? 8 (2r 8)
Substitute values.
Simplify.
400 ? 16r 64
Subtract 64 from each side.
336 ? 16r
Divide each side by 16.
21 ? r
?So, the radius of the tank is about 21 feet.
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Ex. 4 Finding Segment Lengths
(BA)2 BC BD
Use Theorem 10.17
(5)2 x (x 4)
Substitute values.
Simplify.
25 x2 4x
Write in standard form.
0 x2 4x - 25
Use Quadratic Formula.
x
x
Simplify.
Use the positive solution because lengths cannot
be negative. So, x -2 ? 3.39.
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Reminders

• Quiz after this section either Thursday or
  Friday.
• Test will be after 10.7 next week probably
  Tuesday or Wednesday.
• Chapter 10 Algebra Review can be done for Extra
  Credit. Show all work!!!