Question 1053446
any line perpendicular to the line
{{{ y = (3/4)*x - 2 }}} will have slope=
{{{ -1/m = -1/(3/4) }}}
{{{ -1/m = -4/3 }}}
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The equation of the line through (4,-4)
and perpendicular to the given line is:
{{{ ( y - (-4) ) / ( x - 4 ) = -4/3 }}}
{{{ y + 4 = (-4/3)*( x - 4 ) }}}
{{{ y = (-4/3)*x + 16/3 - 4 }}}
{{{ y = (-4/3)*x + 4/3 }}}
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Find the point of intersection of the 2 lines
{{{ (3/4)*x - 2 = (-4/3)*x + 4/3 }}}
Multiply both sides by {{{ 12 }}}
{{{ 9x - 24 = -16x + 16 }}}
{{{ 25x = 40 }}}
{{{ x = 8/5 }}}
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Plug this result back into the original equation
{{{ y = (3/4)*x - 2 }}}
{{{ y = (3/4)*(8/5) - 10/5 }}}
{{{ y = 6/5 - 10/5 }}}
{{{ y = -4/5 }}}
The intersection is at ( 8/5, -4/5 )
Make this the midpoint of the reflected points
{{{ ( x + 4 ) / 2 = 8/5 }}}
{{{ 5*( x + 4 ) = 2*8 }}}
{{{ 5x + 20 = 16 }}}
{{{ 5x = -4 }}}
{{{ x = -4/5 }}}
and
{{{ ( y +(-4) ) / 2 = -4/5 }}}
{{{ ( y - 4 ) / 2 = -4/5 }}}
{{{ 5*( y - 4 ) = -8 }}}
{{{ 5y - 20 = -8 }}}
{{{ 5y = 12 }}}
{{{ y = 12/5 }}}
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The reflected point is ( -4/5, 12/5 )
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Here's the plot of the 2 lines:
{{{ graph( 500, 500, -10, 10, -10, 10, (3/4)*x - 2, -(4/3)*x + 4/3 ) }}}
Looks like my answer can be correct