Consider quadrilateral 𝐴𝐵𝐶𝐷 with vertices 𝐴: negative two, four; 𝐵: negative four, four; 𝐶: negative one, negative five; and 𝐷: one, negative five. Using the distance formula, determine whether the quadrilateral is a parallelogram.
So we’ve been given the coordinates of all four vertices of a quadrilateral and asked to determine whether it’s a parallelogram by using the distance formula. The distance formula enables us to calculate the distance between two points on a coordinate grid.
If those points have coordinates 𝑥 one, 𝑦 one and 𝑥 two, 𝑦 two, then the distance between them is the square root of 𝑥 two minus 𝑥 one all squared plus 𝑦 two minus 𝑦 one all squared. So we can apply the distance formula in order to calculate the lengths of all four sides of the quadrilateral. But how will this help us with determining whether or not it’s a parallelogram?
Well, a key fact about parallelograms is that opposite pairs of sides are congruent. So by considering whether the opposite sides are the same length, we’ll be able to determine whether or not this quadrilateral is a parallelogram.
So we have four lengths that we need to calculate. Let’s begin with the side 𝐴𝐵. The distance formula gives negative four minus negative two squared plus four minus four squared. This becomes the square root of negative two squared, which is the square root of four, which is exactly two. So we found the first length. 𝐴𝐵 is equal to two.
Next, let’s consider the side 𝐵𝐶. The distance formula gives the square root of negative one minus negative four all squared plus negative five minus four all squared. This gives the square root of three squared plus negative nine squared. Three squared is nine. And negative nine squared is 81. So we have the square root of 90, which as a simplified surd is three root 10.
Next, we apply the distance formula for the side 𝐶𝐷. We have the square root of one minus negative one all squared plus negative five minus negative five all squared. This simplifies to the square root of two squared, which is the square root of four, which is just two.
Now we can see at this stage that we do have at least one pair of opposite sides which are congruent. 𝐴𝐵 and 𝐶𝐷 are both two. This doesn’t mean that the quadrilateral is a parallelogram as only one pair of congruent opposite sides would mean that it’s an isosceles trapezoid.
We need to calculate the final length. The length of 𝐷𝐴 is negative two minus one squared plus four minus negative five squared. This is equal to the square root of negative three squared plus nine squared. Negative three squared is nine. And nine squared is 81. So we have the square root of 90, which simplifies to three root 10.
So now we can see that we have a second pair of opposite congruent sides. 𝐵𝐶 and 𝐷𝐴 are both equal to three root 10. So as we’ve shown that this quadrilateral has two pairs of opposite congruent sides, we can conclude that yes it is a parallelogram.