# Do the lines 2x + 4y = 7 , 6y + 3x = 1, 2x – y + 2 = 0 and 6x + 3y = 0, form a rectangle?

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We have the equations of four lines given to us. Now looking at the equations:

2x + 4y = 7

=> 4y = 7 – 2x

=> y = 7/4** – 1/2x**

6y + 3x = 1

=> 6y = 1 - 3x

=> y = 1/6 **– 1/2x**

These two lines have the same slope -1/2 and are parallel.

2x – y + 2 = 0

=> y = **2x** – 2

This has a slope 2 which is the negative inverse of -1/2 and hence the line is perpendicular the first two.

6x + 3y = 0

=> y = -6/3x = **-2x**

Here the slope is -2. This line is not perpendicular to the first two.

**Therefore the four lines do not form a rectangle.**

Given the equation of the lines:

2x + 4y = 7

6y + 3x = 1

2x - y+2 = 0

6x + 3y = 0

First we will rewrite into the slope form ( y= mx + b)

==> 2x + 4y = 7 ==> y1 = (-1/2)x + 7/4

==> 6y + 3x = 1 ==> y2= (-1/2)x + 1/6

==> 2x-y+2 = 0 ==> y3 = 2x +2

==> 6x + 3y = 0 ==> y4= -2x

We notice that:

y1 and y1 are parallel because they have the same slope.

But y3 and y4 are not parallel ( slopes are not equal).

**Then the lines do NOT form a rectangle.**