Question Video: Determining Whether a Given Quadrilateral Is a Parallelogram Using the Slope Formula | Nagwa Question Video: Determining Whether a Given Quadrilateral Is a Parallelogram Using the Slope Formula | Nagwa

Question Video: Determining Whether a Given Quadrilateral Is a Parallelogram Using the Slope Formula

The points πΎ(β5, 0), πΏ(β3, β1), π(β2, 5) and π(β4, 6) are the vertices of quadrilateral πΎπΏππ. Using the slope formula, is the quadrilateral a parallelogram?

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Video Transcript

The points πΎ negative five, zero; πΏ negative three, negative one; π negative two, five; and π negative four, six are the vertices of quadrilateral πΎπΏππ. Using the slope formula, is the quadrilateral a parallelogram?

So weβve been given the coordinates of the four vertices of a quadrilateral and asked to determine whether or not this quadrilateral is a parallelogram. Weβre also told how to do this using the slope formula. Letβs recall its definition. The slope of the line joining the points with coordinates π₯ one, π¦ one and π₯ two π¦, two can be found by calculating the change in π¦ divided by the change in π₯. π¦ two minus π¦ one over π₯ two minus π₯ one.

How would this help us with answering the question? Well, in order for a quadrilateral to be a parallelogram, it needs to have two pairs of parallel sides β each of its pairs of opposite sides. In the case of a parallelogram drawn on a coordinate grid as we have here, this means that the slopes of opposite sides need to be the same. So what we need to do is calculate the slope of each side.

Letβs begin first of all with the side πΎπΏ. π¦ two minus π¦ one is negative one minus zero and π₯ two minus π₯ one is negative three minus negative five. This gives the slope of πΎπΏ as negative a half.

Next, letβs find the slope of the side πΏπ. The change in π¦ is five minus negative one and the change in π₯ is negative two minus negative three. This simplifies to six over one, which is just six. Now, these are a pair of adjacent sides. So weβre not expecting them to have the same slope. If indeed, this quadrilateral is a parallelogram.

Letβs consider the final two sides. Firstly, side ππ, the slope is six minus five over negative four minus negative two. This simplifies to one over negative two, which is better written as negative a half.

Now, notice this is the same as the slope weβve already calculated for the side πΎπΏ. So it is true that the lines πΎπΏ and ππ are parallel. Therefore, our quadrilateral does have at least one pair of parallel sides.

Letβs consider the slope of the final side ππΎ. The slope is zero minus six over negative five minus negative four. This simplifies to negative six over negative one, which is just equal to six. Again, notice that this is the same as the slope of already calculated for the side πΏπ. So now we know that πΏπ is parallel to ππΎ.

So weβve shown that the quadrilateral πΎπΏππ has two pairs of parallel sides. And therefore, our answer to the question βis it a parallelogram?β is yes.

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