# How could you find the area of a parallelogram with the given vertices (without having to graph it)? M(-6, -1) N(-5, 0) P(1, 0) Q(0, -1)

Mar 12, 2016

You need to find the height and base of the parallelogram then put it into the formula: $A = b \left(h\right)$ which gives you the area of the parallelogram.

#### Explanation:

I'm going to preface this by saying that it would help immensely to graph this figure however you don't have to. Before I explain how to go about solving for area lets review the 4 major properties of parallelograms:

1. Opposite sides are parallel to each other

2. Opposite sides are of equal length

3. Opposite angles are equal

4. Diagonals from each point (the corners of the parallelogram) bisect each other at a midpoint

Okay, now that's out of the way we can find the area of your parallelogram.

We need to try to visualize the parallelogram (again, it makes solving much easier when you draw it out). Point M (-6, -1) is below point N (-5, 0), so line MN is the farthest right parameter of the parallelogram. Point N (-5, 0) is to the left of point P (1, 0), line NP then is the top (or roof, if you will) of the parallelogram. Point P (1, 0) is situated above point Q (0, -1), line PQ forms the rightmost line of the parallelogram, the bottom line of the parallelogram is formed by line QM where point Q is adjoined with point M. That is our parallelogram, now to solve for the area.

When it comes to parallelograms since opposite sides are equal we only need to solve for one side each either a right or left side and a top or bottom (you can choose any of the two points as long as they create a line).

For the top and bottom lines their individual lengths are:

(Note: I'm choosing line NP but you could use QM)

You can simply subtract to find the length of the line since these two points are completely linear.

Line NP= N (-5, 0); P (1,0)

$- 5 - 1 = 6$ so now we know that the length of line $N P = 6$

We do the same for a line on the right or left of the parallelogram:

(Note: I'm using line MN however you could use PQ)

Sadly this isn't as simple, there is a formula for the distance between two points unfortunately it convoluted and annoying, luckily I have a few tricks up my sleeve.

Line MN= M (-6, -1); N (-5, 0)

We can use the Pythagorean Theorem (${a}^{2} + {b}^{2} = {c}^{2}$) to solve.

The distance between -5 and -6 is 1 and the distance between -1 and 0 is -1. We plug these values into the Pythagorean Theorem:

${\left(1\right)}^{2} + {\left(- 1\right)}^{2} = {c}^{2}$

$= 1 + 1 = {c}^{2}$

${c}^{2} = 2$

We take the square root.

$c = \sqrt[_]{2}$

The length of line MN equals $\sqrt[_]{2}$

We're not done yet, almost though, hang in there!

Now we have the values for the entire parameter of the parallelogram, to find the area of a parallelogram we use the formula: $A = b \left(h\right)$

b stands for base, which is our long line NP (6)

h stands for height which is our short line MN ($\sqrt[_]{2}$)

So,

$A = 6 \left(\sqrt[_]{2}\right)$

The entire area of your parallelogram is $6 \sqrt[_]{2}$.

Despite the fact that this was really long I hope this helped!