I'll tell you what your mistake was.
You calculated line K's slope as -infinity and for line m, though it's the same point, you might not have recognized it in your hurry and solved for slope consider the same point as twice, giving you a +infinity as slope. So obviously, they seem perpendicular but actually, product of slopes of perpendicular intersecting lines is supposed to be -1, and we are not sure if product of two infinities is 1.
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