The points 𝐾 negative five, zero; 𝐿 negative three, negative one; 𝑀 negative two, five; and 𝑁 negative four, six are the vertices of quadrilateral 𝐾𝐿𝑀𝑁. Using the slope formula, is the quadrilateral a parallelogram?
So we’ve been given the coordinates of the four vertices of a quadrilateral and asked to determine whether or not this quadrilateral is a parallelogram. We’re also told how to do this using the slope formula. Let’s recall its definition. The slope of the line joining the points with coordinates 𝑥 one, 𝑦 one and 𝑥 two 𝑦, two can be found by calculating the change in 𝑦 divided by the change in 𝑥. 𝑦 two minus 𝑦 one over 𝑥 two minus 𝑥 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 — 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.
Let’s begin first of all with the side 𝐾𝐿. 𝑦 two minus 𝑦 one is negative one minus zero and 𝑥 two minus 𝑥 one is negative three minus negative five. This gives the slope of 𝐾𝐿 as negative a half.
Next, let’s find the slope of the side 𝐿𝑀. The change in 𝑦 is five minus negative one and the change in 𝑥 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 we’re not expecting them to have the same slope. If indeed, this quadrilateral is a parallelogram.
Let’s consider the final two sides. Firstly, side 𝑀𝑁, 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 we’ve already calculated for the side 𝐾𝐿. So it is true that the lines 𝐾𝐿 and 𝑀𝑁 are parallel. Therefore, our quadrilateral does have at least one pair of parallel sides.
Let’s consider the slope of the final side 𝑁𝐾. 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 𝐿𝑀. So now we know that 𝐿𝑀 is parallel to 𝑁𝐾.
So we’ve shown that the quadrilateral 𝐾𝐿𝑀𝑁 has two pairs of parallel sides. And therefore, our answer to the question “is it a parallelogram?” is yes.