To find the direct distance between the lizard and the cactus (x), let’s first establish the relationship between the lizard’s position, the cactus's position, and the point of tangency.
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Understand the Geometry:
- The cactus has a diameter of 12 inches, which gives it a radius (r) of: \[ r = \frac{diameter}{2} = \frac{12 \text{ inches}}{2} = 6 \text{ inches} \]
- The distance from the point of tangency on the cactus to the lizard is given as 8 inches.
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Setting Up the Relationship:
- Let (C) be the center of the cactus and (T) be the point of tangency. The radius (CT) from the center to the point of tangency is 6 inches.
- Let (L) be the location of the lizard. The distance from the lizard to the point of tangency (T) is given as 8 inches.
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Applying the Pythagorean Theorem:
- The direct distance (x) from the lizard (L) to the center of the cactus (C) can be visualized as a right triangle formed by the following points:
- From the center of the cactus (C) to the point of tangency (T) as one leg (which is the radius = 6 inches).
- From the point of tangency (T) to the lizard (L) as the other leg (8 inches).
- The direct distance (x) from the lizard (L) to the center (C) is the hypotenuse.
- The direct distance (x) from the lizard (L) to the center of the cactus (C) can be visualized as a right triangle formed by the following points:
Using the Pythagorean theorem:
\[ x^2 = CT^2 + TL^2 \]
Substituting the known values:
\[ x^2 = 6^2 + 8^2 \] \[ x^2 = 36 + 64 \] \[ x^2 = 100 \]
Thus,
\[ x = \sqrt{100} = 10 \text{ inches} \]
- Conclusion: The direct distance between the lizard and the cactus is \( \boxed{10} \) inches.
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