Question 1183029
Find the area of a parallelogram bounded by the y-axis, the line
{{{x = 4}}},

the line

{{{f(x) = 4 + 2x}}}

and the line parallel to {{{f(x)}}} passing through ({{{2}}}, {{{6}}}).


The line parallel to {{{y=4 + 2x}}} will be of the form 

{{{y=2x+k}}}

As it passes through ({{{2}}}, {{{6}}}), we have 

{{{6=2*2+k}}}

{{{k=2}}}

and hence two parallel lines have equations 

{{{y=2x+4}}}
 and
{{{ y=2x+2}}}
.
As the difference in y-intercepts is {{{2}}},the side of parallelogram along y-axis is{{{ 2}}}
.
Further, two other parallel lines are {{{x=0}}} and {{{x=4 }}}and hence vertical distance between them is {{{4}}}


the vertices of the parallelogram are: 

A({{{0}}},{{{4}}}), B({{{0}}},{{{2}}}), 
C({{{4}}},{{{12}}}), and D({{{4}}},{{{10}}})

The edges AB and CD can be considered the bases; then the length of the bases is {{{2}}} and the height is {{{4 }}}(the horizontal distance between AB and CD).

The area of a parallelogram is base times height:

 {{{A=2*4=8 }}}units


{{{ drawing ( 600, 600, -10, 15, -10, 15,
circle(0,4,.12), locate(0,4,A), circle(0,2,.12), locate(0,2,B),circle(4,12,.12), locate(4,12,C),circle(4,10,.12), locate(4,10,D),
circle(2,6,.12), locate(2,6,p(2,6)),
green(line(0.1,15,0.1,-10)),green(line(4,15,4,-10)),

blue(line(0.1,4,0.1,2)),blue(line(4,12,4,10)),
blue(line(0.1,4,4,12)), blue(line(0.1,2,4,10)),
graph( 600, 600, -10, 15, -10, 15, 2x+2, 2x+4)) }}}