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10.3 Chords and Segment Relationships

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1. The document discusses properties of chords, secants, and tangents of circles, including: if a radius or diameter is perpendicular to a chord it bisects the chord and arc; if one chord is a perpendicular bisector of another it is a diameter; and the chord-chord, secant-secant, and secant-tangent products. 2. Examples are provided to demonstrate finding lengths and values using these properties, such as finding the diameter or value of x given lengths of line segments. 3. The document also describes how the property of a perpendicular bisector being a diameter was used to find the center of a circle by measuring the midpoint.

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Chords and Segment Relationships The student is able to (I can): Apply properties of chords Find the lengths of segments formed by lines that intersect circles
In the same circle (or in congruent circles), two minor arcs are congruent if and only if their corresponding chords are congruent. E R O F m m FO RE FO RE
If a radius or diameter is perpendicular to a chord, then it bisects the chord and its arc. ER GO G E O R A GA AO GR RO
Example Find the length of . BU B L U E 2 5 x
Example Find the length of . BU B L U E 3 2 5 2 2 2 3 5 x + = x x = 4 BU = 2(4) = 8
If one chord of a circle is a perpendicular bisector of another chord, then the first chord is a diameter. The midpoint of the diameter would be the center of the circle. G W O N A If is a bisector of , then is a diameter. WN GO WN
A while back, I needed to find the center of a circle, so I used this property.
A while back, I needed to find the center of a circle, so I used this property. Step 1: I drew a chord Step 2: I drew the perpendicular bisector, which is a diameter.
A while back, I needed to find the center of a circle, so I used this property. Step 3: I measured and found the midpoint of the diameter, which gave me the center!
chord-chord product if two chords intersect in the interior of a circle, then the products of the lengths of the segments of the chords are equal. S P A C E P S E A AC A A =
Examples 1. Find the value of x. 2. What is the diameter of the circle? 9 12 x 6 ( ) 9 6 12 x = 9 72 x = 8 x = ( ) 4 6 6 x = 4 36 x = 9 x = diameter 4 9 13 = + = 6 6 4 x
secant-secant product if two secants intersect in the exterior of a circle, then the product of the lengths of one secant segment and its external segment equals the product of the lengths of the other secant segment and its external segment. (whole outside = whole outside) O P M E T M T O M EM M P =
Example Find the value of x. 10 x 8 12 ( ) ( ) 10 10 12 20 10 100 240 10 140 14 x x x x + = + = = =
secant-tangent product if a secant and a tangent intersect in the exterior of a circle, then the product of the lengths of the secant segment and its external segment equals the length of the tangent segment squared. (whole outside = tangent2) G O R F 2 FO RO GO =
Example Find the value of x. 8 10 x ( ) ( )( ) 2 2 2 10 8 8 18 8 144 12 x x x x + = = = =

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10.3 Chords and Segment Relationships

  • 1. Chords and Segment Relationships The student is able to (I can): Apply properties of chords Find the lengths of segments formed by lines that intersect circles
  • 2. In the same circle (or in congruent circles), two minor arcs are congruent if and only if their corresponding chords are congruent. E R O F m m FO RE FO RE
  • 3. If a radius or diameter is perpendicular to a chord, then it bisects the chord and its arc. ER GO G E O R A GA AO GR RO
  • 4. Example Find the length of . BU B L U E 2 5 x
  • 5. Example Find the length of . BU B L U E 3 2 5 2 2 2 3 5 x + = x x = 4 BU = 2(4) = 8
  • 6. If one chord of a circle is a perpendicular bisector of another chord, then the first chord is a diameter. The midpoint of the diameter would be the center of the circle. G W O N A If is a bisector of , then is a diameter. WN GO WN
  • 7. A while back, I needed to find the center of a circle, so I used this property.
  • 8. A while back, I needed to find the center of a circle, so I used this property. Step 1: I drew a chord Step 2: I drew the perpendicular bisector, which is a diameter.
  • 9. A while back, I needed to find the center of a circle, so I used this property. Step 3: I measured and found the midpoint of the diameter, which gave me the center!
  • 10. chord-chord product if two chords intersect in the interior of a circle, then the products of the lengths of the segments of the chords are equal. S P A C E P S E A AC A A =
  • 11. Examples 1. Find the value of x. 2. What is the diameter of the circle? 9 12 x 6 ( ) 9 6 12 x = 9 72 x = 8 x = ( ) 4 6 6 x = 4 36 x = 9 x = diameter 4 9 13 = + = 6 6 4 x
  • 12. secant-secant product if two secants intersect in the exterior of a circle, then the product of the lengths of one secant segment and its external segment equals the product of the lengths of the other secant segment and its external segment. (whole outside = whole outside) O P M E T M T O M EM M P =
  • 13. Example Find the value of x. 10 x 8 12 ( ) ( ) 10 10 12 20 10 100 240 10 140 14 x x x x + = + = = =
  • 14. secant-tangent product if a secant and a tangent intersect in the exterior of a circle, then the product of the lengths of the secant segment and its external segment equals the length of the tangent segment squared. (whole outside = tangent2) G O R F 2 FO RO GO =
  • 15. Example Find the value of x. 8 10 x ( ) ( )( ) 2 2 2 10 8 8 18 8 144 12 x x x x + = = = =