Question

9) Determine which sides, if any, of the figures are parallel, perpendicular, or neither.

Rectangle BACD has coordinate B(-4,-3), A(-1, -7),C(3,-4), and D(0,0).

Answer 1

Parallel: AC and DB, BA and CD

Perpendicular: AC and CD, DB and CD, DB and BA, BA and AC

Neither: None

Explanation:

Step 1: Find Slopes

Let's first find the slopes of each side of the rectangle, as that will helps us determine which sides are parallel, perpendicular, or neither.

Recall that the formula for finding the slope between two points is
`(y_2 - y_1)/(x_2-x_1)` where
`(x_1, y_1)` and
`(x_2,y_2)` are the coordinates of the two points. To avoid confusion, I will be taking the first point I list as
`(x_1,y_1)` and the second point as
`(x_2,y_2)`.

Slope of
`BA`:

`(-7-(-3))/(-1-(-4))\\=(-7+3)/(-1+4) \\= -(4)/(3)`

Slope of
`AC`:

`(-4-(-7))/(3-(-1)) \\=(-4+7)/(3+1) \\=(3)/(4)`

Slope of
`CD`:

`(0-(-4))/(0-3) \\=(4)/(-3) \\=-(4)/(3)`

Slope of
`DB`:

`(-3-0)/(-4-0) \\=(-3)/(-4) \\=(3)/(4)`

Step 2: Determine which sides are parallel, perpendicular, or neither

Now that we found the slopes of the sides, we can determine which sides are parallel, perpendicular, or neither.

Recall that parallel lines have the same slope.
`\bf AC` and
`\bf DB`, along with
`\bf BA`and
`\bf CD`, have the same slope, so they are parallel. No other pair of sides has the same slope, so these are our only parallel pairs.

For two lines to be perpendicular, the product of their slopes must be
`-1`.
`\bf AC` and
`\bf CD`,
`\bf DB` and
`\bf CD`,
`\bf DB` and
`\bf BA`, and
`\bf BA` and
`\bf AC`
`\bf`meet that criteria, so they are perpendicular. No other pair of sides meets the criteria, so these are our only perpendicular pairs. Hope this helps!