Sec. 12 – 4  Measures & Segment Lengths in Circles Objectives: 1) To find the measures of  s formed by chords, secants, & tangents. 2) To find the lengths of segments associated with circles.

Secants E A B F Secant – A line that intersects a circle in exactly 2 points. EF or AB are secants AB is a chord

(Thm 11 – 11) The measure of an  formed by 2 lines that intersect inside a circle is (Thm 11 – 11) The measure of an  formed by 2 lines that intersect inside a circle is m  1 = ½(x + y) Measure of intercepted arcs 1 x°x° y°y°

(…Thm 11 – 11 Continues) The measure of an  formed by 2 lines that intersect outside a circle is (…Thm 11 – 11 Continues) The measure of an  formed by 2 lines that intersect outside a circle is m  1 = ½(x - y) Smaller Arc Larger Arc x°x° y°y° 1 x°x° y°y° 1 2 Secants: x°x° y°y° 1 Tangent & a Secant 2 Tangents 3 cases:

Ex.1 & 2: Find the measure of arc x. Find the measure of arc x. Find the m  x. Find the m  x. 94° 112° x°x° m  1 = ½(x + y) 94 = ½(112 + x) 188 = (112 + x) 76° = x 68° 104° 92° 268° x°x° m  x = ½(x - y) m  x = ½( ) m  x = ½(176) m  x = 88°

Thm (11 – 12) Lengths of Secants, Tangents, & Chords 2 Chords ac b d ab = cd 2 Secants x w z y w(w + x) = y(y + z) Tangent & Secant t y z t 2 = y(y + z)

Ex. 3 & 4 Find length of x. Find length of x. Find the length of g. Find the length of g. 3x 7 5 ab = cd (3)(7) = (x)(5) 21 = 5x 4.2 = x 15 8 g t 2 = y(y + z) 15 2 = 8(8 + g) 225 = g 161 = 8g = g

Ex.5: 2 Secants Find the length of x. Find the length of x x w(w + x) = y(y + z) 14( ) = 16(16 + x) (34)(14) = x 476 = x 220 = 16x 3.75 = x

Ex.6: A little bit of everything! Find the measures of the missing variables Find the measures of the missing variables 9 12 k 8 a°a° r 60° 175° Solve for k first. w(w + x) = y(y + z) 9(9 + 12) = 8(8 + k) 186 = k k = 15.6 Next solve for r t 2 = y(y + z) r 2 = 8( ) r 2 = 189 r = 13.7 Lastly solve for m  a m  1 = ½(x - y) m  a = ½(175 – 60) m  a = 57.5°

What have we learned?? When dealing with angle measures formed by intersecting secants or tangents you either add or subtract the intercepted arcs depending on where the lines intersect. When dealing with angle measures formed by intersecting secants or tangents you either add or subtract the intercepted arcs depending on where the lines intersect. There are 3 formulas to solve for segments lengths inside of circles, it depends on which segments you are dealing with: Secants, Chords, or Tangents. There are 3 formulas to solve for segments lengths inside of circles, it depends on which segments you are dealing with: Secants, Chords, or Tangents.