Circles Geometry: Distance for Three Common Tangents
To determine the minimum distance between the centers of two circles that have exactly three common tangents, we need to understand the relationship between the distance ($d$) between their centers and their radii ($r_1$ and $r_2$).
Condition for Three Common Tangents
Exactly three common tangents exist between two circles when they touch each other externally.
- When circles touch externally, the distance ($d$) between their centers is equal to the sum of their radii ($r_1 + r_2$).
Calculating Minimum Distance
Given:
- Radius of the first circle, $r_1 = 5$ cm
- Radius of the second circle, $r_2 = 3$ cm
For the circles to touch externally, the distance ($d$) between their centers must be:
$d = r_1 + r_2$
Substituting the given radii:
$d = 5 \text{ cm} + 3 \text{ cm}$
$d = 8 \text{ cm}$
Therefore, the minimum distance between the centers of the two circles is 8 cm.
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