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Understanding Segment Lengths in Circles: Secants, Chords, and Tangents

This guide explores the lengths of segments associated with circles, specifically secants, chords, and tangents. Learn how to use key formulas to find segment lengths, including relationships between secants and chords. Through examples, you'll understand how to apply the equations to solve for unknown lengths, using the products of the segments created by the secants and tangents. The guide also emphasizes the importance of understanding angle measures formed by these intersecting lines and provides practice problems to enhance your skills in geometric calculations related to circles.

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Understanding Segment Lengths in Circles: Secants, Chords, and Tangents

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  1. Segment Lengths in Circles Objectives: To find the lengths of segments associated with circles.

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

  3. Lengths of Secants, Tangents, & Chords 2 Secants 2 Chords Tangent & Secant y a c t z x b z d w y a•b = c•d t2 = y(y + z) w(w + x) = y(y + z)

  4. Find length of x. Find the length of g. Examples: 8 15 g 3 x 7 5 t2 = y(y + z) 152 = 8(8 + g) 225 = 64 + 8g 161 = 8g 20.125 = g a•b = c•d (3)•(7) = (x)•(5) 21 = 5x 4.2 = x

  5. Ex.: 2 Secants Find the length of x. 20 14 16 w(w + x) = y(y + z) 14(14 + 20) = 16(16 + x) (34)(14) = 256 + 16x 476 = 256 + 16x 220 = 16x 3.75 = x x

  6. Ex. : A little bit of everything! Find the measures of the missing variables Solve for k first. w(w + x) = y(y + z) 9(9 + 12) = 8(8 + k) 186 = 64 + 8k k = 15.6 12 k 175° 9 8 60° Next solve for r t2 = y(y + z) r2 = 8(8 + 15.6) r2 = 189 r = 13.7 a° r Lastly solve for ma m1 = ½(x - y) ma = ½(175 – 60) ma = 57.5°

  7. 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. • 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.

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