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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  • 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?     ### 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.

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

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

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

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

• • • • • » 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.

• • •  The measure of the common internal tangent of the two circles is 93 units. (ノ^_^)ノ