How do you find the perpendicular distance between the straight lines $y=2x+7$ and $y=2x+2$ ?

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Answer 1 · 2015-11-30T15:31:35

$5/ sqrt 5 = sqrt 5$

$r : y = 2x + 7$

$s : y = 2x + 2$

$r$ and $s$ are parallel, and both are perpendicular to

$t : y = -1/2 x + b$

Let $b = 0$ and find $r cap t = P, s cap t = Q$

$2x + n = -1/2 x$

$4x + 2n = -x$

$x = - (2n) / 5$

$y = -1/2 x = n/5$

$|PQ|^2 = (-(2n_1)/5 + (2n_2)/5)^2 + ((n_1)/5 - (n_2)/5)^2$

$|PQ|^2 = 4/25 (n_2 - n_1)^2 + 1/25 (n_1 - n_2)^2$

$|PQ| = sqrt {1/5} |n_1 - n_2|$

$n_1 = 7, n_2 = 2$

How do you find the perpendicular distance between the straight lines y=2x+7y=2x+7 and y=2x+2y=2x+2?