Question 382595
Solution 1: The coordinates of N are (x, 0).  Use the Pythagorean Theorem.
{{{(BA)^2 + (BN)^2 = (AN)^2}}};
{{{(3-0)^2 + (4-0)^2 + (3-x)^2 + (4-0)^2 = (x-0)^2 + (0-0)^2}}};
{{{9 + 16 + 9-6x+ x^2 + 16 = x^2}}};
50 - 6x = 0;
50 = 6x;
x = 25/3.



Solution2: Slope of line AB  is 4/3.  Thus the slope of line BN must be the negative reciprocal, or -3/4.
Hence {{{(4-0)/(3-x) = -3/4}}}, or 16 = -9 + 3x, or 25 = 3x, or x = 25/3.