Angle Measures and Segment Lengths in Circles Objectives

Angle Measures and 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.

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

l Theorem. The measure of an formed by 2 lines that intersect inside a circle is half of the arcs measures, intercepted by the lines. Arcs are across from angles. m 1 = ½(x + y) Measure of intercepted arcs y° 1 x°

l Theorem. The measure of an formed by 2 lines that intersect outside a circle is m 1 = ½(x - y) 3 cases: 2 Secants: Smaller Arc Larger Arc 1 1 Tangent & a Secant 1 y° y° y° x° x° x° 2 Tangents

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

Lengths of Secants, Tangents, & Chords 2 Secants 2 Chords Tangent & Secant y a c t x b z z d w y a • b = c • d t 2 = y(y + z) w(w + x) = y(y + z)

Ex. 3 & 4 l Find length of x. l Find the length of g. 8 3 15 x g 7 5 a • b = c • d (3) • (7) = (x) • (5) 21 = 5 x 4. 2 = x t 2 = y(y + z) 152 = 8(8 + g) 225 = 64 + 8 g 161 = 8 g 20. 125 = g

Ex. 5: 2 Secants l Find the length of x. 20 14 w(w + x) = y(y + z) 14(14 + 20) = 16(16 + x) (34)(14) = 256 + 16 x 476 = 256 + 16 x 220 = 16 x 3. 75 = x 16 x

Ex. 6: A little bit of everything! l Find the measures of the missing variables Solve for k first. w(w + x) = y(y + z) 9(9 + 12) = 8(8 + k) 12 k 9 a° 175° 186 = 64 + 8 k k = 15. 6 8 60° r Lastly solve for m a m 1 = ½(x - y) m a = ½(175 – 60) m a = 57. 5° Next solve for r t 2 = y(y + z) r 2 = 8(8 + 15. 6) r 2 = 189 r = 13. 7

What have we learned? ? l 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. l 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.