Question 1011678
(FIXED EARLIER MISTAKE)


Place the triangle on a cartesian system with the right angle of the room's corner at the origin.  The railing fits on a line,  {{{y=-(3/4)x-9}}}.  The segment length labeled, "x" , is on the line {{{y=(4/3)x}}}, maybe now a bad choice of naming for variables, perpendicular to the railing.  


Find the other endpoint of "x"; which is the intersection point of  {{{y=-(3/4)x-9}}} and {{{y=(4/3)x}}}.
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{{{4x/3=-(3/4)x-9}}}
{{{4x/3+(3/4)x=-9}}}
Multiply L&R by 12.
{{{16x+9x=-9*12}}}
{{{25x=-72}}}
{{{highlight_green(x=-72/25)}}}
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{{{y=(4/3)(-72/25)}}}
{{{y=(-4*24)/25}}}
{{{highlight_green(y=-96/25)}}}
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Intersection point is  ( -72/25, -96/25 ).

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Final task is USE THE DISTANCE FORMULA to find distance between  (0,0) and ( -72/25, -96/25 ).