The given line is x/1 = y/0 = z/0. This represents the x-axis, where any point on the line is of the form (k, 0, 0). The given point is P(2, 5, 7).
Let the foot of the perpendicular from P(2, 5, 7) to the line be M(k, 0, 0). The direction vector of the line is d = (1, 0, 0). The vector PM is (k-2, -5, -7). Since PM is perpendicular to the line, their dot product is zero:
$$(k-2)(1) + (-5)(0) + (-7)(0) = 0$$This gives k - 2 = 0, so k = 2. Thus, the foot of the perpendicular is M(2, 0, 0).
The length of the perpendicular is the distance between P(2, 5, 7) and M(2, 0, 0):
$$d = \sqrt{(2-2)^2 + (5-0)^2 + (7-0)^2}$$ $$d = \sqrt{0^2 + 5^2 + 7^2} = \sqrt{25 + 49} = \sqrt{74}$$Final Answer: \sqrt{74}
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