### Video Transcript

Consider quadrilateral ๐ด๐ต๐ถ๐ท with vertices ๐ด: negative two, four; ๐ต: negative four, four; ๐ถ: negative one, negative five; and ๐ท: one, negative five. Using the distance formula, determine whether the quadrilateral is a parallelogram.

So weโve been given the coordinates of all four vertices of a quadrilateral and asked to determine whether itโs 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 ๐ฅ one, ๐ฆ one and ๐ฅ two, ๐ฆ two, then the distance between them is the square root of ๐ฅ two minus ๐ฅ one all squared plus ๐ฆ two minus ๐ฆ 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 itโs 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, weโll be able to determine whether or not this quadrilateral is a parallelogram.

So we have four lengths that we need to calculate. Letโs begin with the side ๐ด๐ต. 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. ๐ด๐ต is equal to two.

Next, letโs consider the side ๐ต๐ถ. 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 ๐ถ๐ท. 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. ๐ด๐ต and ๐ถ๐ท are both two. This doesnโt mean that the quadrilateral is a parallelogram as only one pair of congruent opposite sides would mean that itโs an isosceles trapezoid.

We need to calculate the final length. The length of ๐ท๐ด 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. ๐ต๐ถ and ๐ท๐ด are both equal to three root 10. So as weโve shown that this quadrilateral has two pairs of opposite congruent sides, we can conclude that yes it is a parallelogram.