Tangent Circles and Segment Lengths

Finding Segment Lengths in Tangent Circles

Problem 5

  • Problem Statement: Find the indicated segment length, assuming tangent lines are tangent.
  • Given Values: Two external tangent segments with lengths 12 and 16 are provided.
  • Concept: If two tangent segments are drawn to a circle from the same external point, then the tangent segments are congruent.
  • Application: The length of the missing segment is 16, because it is tangent to the circle from the same point

Problem 6

  • Problem Statement: Identify the length of the other tangent.
  • Given Values: Given value of tangent 8.5
  • Concept: The concept of congruent tangent segments from a single point to a circle is used.
  • Application: The length of the missing segment is 8.5 because both tangents from the same point are equivalent.

Problem 7

  • Problem Statement: Calculate segment x.
    • Two secant segments intersect outside the circle.
  • Given Values: The lengths of segments are 1.5, 1, and 2
  • Formulation of Equation:
    • (1.5+1)∗1=(2+x)∗2(1.5 + 1) * 1 = (2 + x) * 2
  • Solving for x:
    • 2.5∗1=4+2x2.5 * 1 = 4 + 2x
    • 2.5=4+2x2.5 = 4 + 2x
    • 2x=−1.52x = -1.5
    • x=−0.75x = -0.75
    • However, since length cannot be negative, there may be an error in copying the values.

Problem 8

  • Problem Statement: Find the value of x, given tangent and secant segments.
  • Given Values: The tangent segment is 6.4, the whole secant segment is x, and the external part of the secant segment is 1.
  • Equation setup
    • (tangent)2=(external secant)∗(whole secant)(tangent)^2 = (external\, secant) * (whole \,secant)
    • (6.4)2=1∗(1+x)(6.4)^2 = 1 * (1 + x)
  • Solve for x
    • 40.96=1+x40.96 = 1 + x
    • x=39.96x = 39.96