Sec. 12 – 4  Measures & Segment Lengths in 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 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 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 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 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 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 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°

Homework: p. 691 #1-6, 9-14, 21, 25