14 5 segment measures lesson

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14 5 segment measures lesson

  1. 1. Sec. 14-5Sec. 14-5 ∠∠ Measures & SegmentMeasures & Segment Lengths in CirclesLengths in Circles Objectives:Objectives: 1) To find the measures of1) To find the measures of ∠∠s formeds formed by chords, secants, & tangents.by chords, secants, & tangents. 2) To find the lengths of segments2) To find the lengths of segments associated with circles.associated with circles.
  2. 2. SecantsSecants E A B F Secant – A line that intersects a circle in exactly 2 points. •EF or AB are secants •AB is a chord
  3. 3.  (Thm 11 – 11) The measure of an(Thm 11 – 11) The measure of an ∠∠ formed by 2 lines that intersectformed by 2 lines that intersect insideinside aa circle iscircle is m∠1 = ½(x + y) Measure of intercepted arcs 1 x° y°
  4. 4.  (…Thm 11 – 11 Continues) The measure(…Thm 11 – 11 Continues) The measure of anof an ∠∠ formed by 2 lines that intersectformed by 2 lines that intersect outsideoutside a circle isa circle is m∠1 = ½(x - y) Smaller Arc Larger Arc x° y° 1 x° y° 1 2 Secants: x° y° 1 Tangent & a Secant 2 Tangents 3 cases:
  5. 5. Ex.1 & 2:Ex.1 & 2:  Find the measure ofFind the measure of arc x.arc x.  Find the mFind the m∠∠x.x. 94° 112° x° m∠1 = ½(x + y) 94 = ½(112 + x) 188 = (112 + x) 76° = x 68° 104° 92° 268° x° m∠x = ½(x - y) m∠x = ½(268 - 92) m∠x = ½(176) m∠x = 88°
  6. 6. Thm (11 – 12) Lengths of Secants,Thm (11 – 12) Lengths of Secants, Tangents, & ChordsTangents, & Chords 2 Chords a c b d a•b = c•d 2 Secants x w z y w(w + x) = y(y + z) Tangent & Secant t y z t2 = y(y + z)
  7. 7. Ex. 3 & 4Ex. 3 & 4  Find length of x.Find length of x.  Find the length of g.Find the length of g. 3 x 7 5 15 8 g
  8. 8. Ex. 3 & 4Ex. 3 & 4  Find length of x.Find length of x.  Find the length of g.Find the length of g. 3 x 7 5 a•b = c•d (3)•(7) = (x)•(5) 21 = 5x 4.2 = x 15 8 g t2 = y(y + z) 152 = 8(8 + g) 225 = 64 + 8g 161 = 8g 20.125 = g
  9. 9. Ex.5: 2 SecantsEx.5: 2 Secants  Find the length of x.Find the length of x. 14 20 16 x w(w + x) = y(y + z) 14(14 + 20) = 16(16 + x) (34)(14) = 256 + 16x 476 = 256 + 16x 220 = 16x 3.75 = x
  10. 10. Ex.6: A little bit of everything!Ex.6: A little bit of everything!  Find the measures of the missing variablesFind the measures of the missing variables 9 12 k 8 a° r 60° 175°
  11. 11. Ex.6: A little bit of everything!Ex.6: A little bit of everything!  Find the measures of the missing variablesFind the measures of the missing variables 9 12 k 8 a° r 60° 175° Solve for k first. w(w + x) = y(y + z) 9(9 + 12) = 8(8 + k) 186 = 64 + 8k k = 15.6 Next solve for r t2 = y(y + z) r2 = 8(8 + 15.6) r2 = 189 r = 13.7 Lastly solve for m∠a m∠1 = ½(x - y) m∠a = ½(175 – 60)
  12. 12. What have we learned??What have we learned??  When dealing with angle measures formed byWhen dealing with angle measures formed by intersecting secants or tangents you either addintersecting secants or tangents you either add or subtract the intercepted arcs depending onor subtract the intercepted arcs depending on where the lines intersect.where the lines intersect.  There are 3 formulas to solve for segmentsThere are 3 formulas to solve for segments lengths inside of circles, it depends on whichlengths inside of circles, it depends on which segments you are dealing with: Secants,segments you are dealing with: Secants, Chords, or Tangents.Chords, or Tangents.

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