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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Krntnicolepadinggo Krntnicolepadinggo

Math

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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?

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KAntoineDoixgo KAntoineDoixgo · Answer 1

✒️CIRCLE

[tex]••••••••••••••••••••••••••••••••••••••••••••••••••[/tex]

[tex] \large\underline{\mathbb{PROBLEM}:} [/tex]

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?

[tex]••••••••••••••••••••••••••••••••••••••••••••••••••[/tex]

[tex] \large\underline{\mathbb{ANSWER}:} [/tex]

[tex] \qquad \LARGE \:\: \rm{9\sqrt3 \: units} [/tex]

[tex]••••••••••••••••••••••••••••••••••••••••••••••••••[/tex]

[tex] \large\underline{\mathbb{SOLUTION}:} [/tex]

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.

[tex] \Delta ADF \: \sim \Delta BCF [/tex]

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

[tex] \frac{\,AD\,}{BC} = \frac{\,AF\,}{B F} \\ [/tex]

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

[tex] \frac{\,3\,}{6} = \frac{x}{\,18-x\,} \\ [/tex]

[tex] \frac{\,1\,}{2} = \frac{x}{\,18-x\,} \\ [/tex]

[tex] 1(18 - x) = 2(x) [/tex]

[tex] 18 - x = 2x [/tex]

[tex] 2x + x = 18 [/tex]

[tex] 3x = 18 [/tex]

[tex] \frac{\,3x\,}{3} = \frac{\,18\,}{3} \\ [/tex]

[tex] x = 6 [/tex]

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

[tex] (DF)^2 + (AD)^2 = (AF)^2 [/tex]

[tex] (DF)^2 + (3)^2 = (6)^2 [/tex]

[tex] (DF)^2 + 9 = 36 [/tex]

[tex] (DF)^2 = 36 - 9 [/tex]

[tex] (DF)^2 = 27 [/tex]

[tex] \sqrt{(DF)^2} = \sqrt{27} [/tex]

[tex] DF = 3\sqrt3 [/tex]

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

[tex] \frac{\,AD\,}{BC} = \frac{\,DF\,}{CF} \\ [/tex]

[tex] \frac{\,3\,}{6} = \frac{\,3\sqrt3\,}{CF} \\ [/tex]

[tex] \frac{\,1\,}{2} = \frac{\,3\sqrt3\,}{CF} \\ [/tex]

[tex] 1(CF) = 2(3\sqrt3) \\ [/tex]

[tex] CF = 6\sqrt3 \\ [/tex]

» 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.

[tex] DC = DF + CF [/tex]

[tex] DC = 3\sqrt3 + 6\sqrt3 [/tex]

[tex] DC = 9\sqrt3 [/tex]

[tex] \therefore [/tex] The measure of the common internal tangent of the two circles is 9√3 units.

[tex]••••••••••••••••••••••••••••••••••••••••••••••••••[/tex]

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