3. The distance between the centers of the two circles in Figure 10, having radii, of 3 and 6, is 18. How long is the common internal tangent?

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3. The distance between the centers of the two circles in Figure 10, having radii


of 3 and 6, is 18. How long is the common internal tangent?​

✒️CIRCLE

 \largenderline{\mathbb{PROBLEM}:}

  • The distance between the centers of the two circles in Figure 10, having radii
  • of 3 and 6, is 18. How long is the common internal tangent?

 \largenderline{\mathbb{ANSWER}:}

 \qquad \LARGE \:\:  \rm{9\sqrt3 \: units}

 \largenderline{\mathbb{SOLUTION}:}

Illustration:

  • Label the points A and B as the centers of the two circles. Draw a segment between them measuring 18 units.
  • Draw one common internal tangent labeled as DC.
  • From their point of tangency (D and C), draw radii AD and BC then indicate their measures.
  • Label intersection of AB and DC as F.

Solving:

» By the construction, we have created two similar right triangles.

  •  \Delta ADF \: \sim \Delta BCF

» And then, we can find the proportion of the sides of these triangles as:

  •  \frac{\,AD\,}{BC} = \frac{\,AF\,}{B F} \\

» Let x be the measure of AF and (18-x) for B F since AB is 18. Find x.

  •  \frac{\,3\,}{6} = \frac{x}{\,18-x\,} \\
  •  \frac{\,1\,}{2} = \frac{x}{\,18-x\,} \\
  •  1(18 - x) = 2(x)
  •  18 - x = 2x
  •  2x + x = 18
  •  3x = 18
  •  \frac{\,3x\,}{3} = \frac{\,18\,}{3} \\
  •  x = 6

» Thus, AF measures 6 units. Solve for DF using the Pythagorean Theorem.

  •  (DF)^2 + (AD)^2 = (AF)^2
  •  (DF)^2 + (3)^2 = (6)^2
  •  (DF)^2 + 9 = 36
  •  (DF)^2 = 36 - 9
  •  (DF)^2 = 27
  •  \sqrt{(DF)^2} = \sqrt{27}
  •  DF = 3\sqrt3

» Thus, DF measures 3√3 units. Find CF using proportion.

  •  \frac{\,AD\,}{BC} = \frac{\,DF\,}{CF} \\
  •  \frac{\,3\,}{6} = \frac{\,3\sqrt3\,}{CF} \\
  •  \frac{\,1\,}{2} = \frac{\,3\sqrt3\,}{CF} \\
  •  1(CF) = 2(3\sqrt3) \\
  •  CF = 6\sqrt3 \\

» Thus, CF measures 6√3 units. Find the measure of the common internal tangent or the measure of DC by the sum of DF and CF.

  •  DC = DF + CF
  •  DC = 3\sqrt3 + 6\sqrt3
  •  DC = 9\sqrt3

 \therefore The measure of the common internal tangent of the two circles is 93 units.

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