Let the distance Janet has walked be \( x \) kilometers. Since Michelle has walked 2 km farther than Janet, the distance Michelle has walked is \( x + 2 \) kilometers.
According to the problem, at the point where they are 10 km apart, we can use the Pythagorean theorem. The distances Janet and Michelle walked form a right triangle where:
- Janet's distance (north) is \( x \)
- Michelle's distance (east) is \( x + 2 \)
- The hypotenuse (the distance between them) is 10 km
Thus, we have the equation:
\[ x^2 + (x + 2)^2 = 10^2 \]
Expanding \((x + 2)^2\):
\[ x^2 + (x^2 + 4x + 4) = 100 \]
Combining the terms:
\[ 2x^2 + 4x + 4 = 100 \]
Subtracting 100 from both sides:
\[ 2x^2 + 4x + 4 - 100 = 0 \]
Simplifying this:
\[ 2x^2 + 4x - 96 = 0 \]
Dividing the entire equation by 2:
\[ x^2 + 2x - 48 = 0 \]
Now, we can factor this quadratic equation:
\[ (x + 8)(x - 6) = 0 \]
Setting each factor to zero gives us:
\[ x + 8 = 0 \quad \Rightarrow \quad x = -8 \quad \text{(not a valid distance)} \] \[ x - 6 = 0 \quad \Rightarrow \quad x = 6 \]
Thus, Janet has walked 6 km. To find out how far Michelle has walked:
\[ \text{Distance Michelle walked} = x + 2 = 6 + 2 = 8 \text{ km} \]
Therefore, the distance Michelle has walked is:
\[ \boxed{8 \text{ km}} \]
To confirm, we check the distances:
- Janet's distance: 6 km
- Michelle's distance: 8 km
Using the Pythagorean theorem:
\[ 6^2 + 8^2 = 36 + 64 = 100 \] \[ \sqrt{100} = 10 \]
This matches the condition given in the problem. Thus, our solution is correct: Michelle has walked \( \boxed{8} \) kilometers.
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