A parallelogram 𝐴𝐵𝐶𝐷 has vertices 𝐴: negative five, five; 𝐵: nine, three; 𝐶: eight, negative four; and 𝐷: negative six, negative two. Firstly, calculate the length of 𝐴𝐶. Give an exact answer. Secondly, calculate the length of 𝐵𝐷. Give an exact answer. Hence, state whether or not the parallelogram is a rectangle.
So we’ve been given the coordinates of the four vertices of a parallelogram and asked to calculate the lengths of 𝐴𝐶 and 𝐵𝐷, which are the diagonals of the parallelogram. To answer this question, we’ll need to use the distance formula, which tells us how to find the distance between two points on a coordinate grid. Let’s recall its definition.
The distance 𝑑 between two points with coordinates 𝑥 one, 𝑦 one and 𝑥 two, 𝑦 two is given by the square root of 𝑥 two minus 𝑥 one squared plus 𝑦 two minus 𝑦 one squared. This is an application of the Pythagorean theorem, where 𝑥 two minus 𝑥 one and 𝑦 two minus 𝑦 one are the horizontal and vertical lengths of a right-angled triangle and 𝑑 is the length of the third side, the hypotenuse.
To find the length of 𝐴𝐶, first of all, we need to substitute the coordinates of 𝐴 and 𝐶 into the distance formula. We have then that 𝑑 one is equal to the square root of eight minus negative five squared plus negative four minus five squared. This is equal to the square root of 13 squared plus negative nine squared. 13 squared is 169. And negative nine squared is 81. So we have the square root of 169 plus 81, which is equal to the square root of 250. We were asked to give an exact answer for the length of 𝐴𝐶. So we’ll leave it in terms of a surd, the square root of 250.
Next, let’s calculate the length of 𝐵𝐷 by substituting the coordinates of the points 𝐵 and 𝐷 into the distance formula. We have that 𝑑 two is equal to the square root of negative six minus nine squared plus negative two minus three squared. This is equal to the square root of negative 15 squared plus negative five squared. Negative 15 squared is equal to 225. And negative five squared is equal to 25. So we have the square root of 225 plus 25, which simplifies to the square root of 250.
So we found the lengths of 𝐴𝐶 and 𝐵𝐷. And you will have noticed, I’m sure, that they are in fact the same. They’re both equal to the square root of 250. How does this help us with answering the final part of the question? Well, remember, 𝐴𝐶 and 𝐵𝐷 are the diagonals of this parallelogram. A key fact about rectangles, which isn’t true of parallelograms in general, is that their diagonals are equal. Therefore, as the diagonals of this parallelogram are equal in length, 𝐴𝐵𝐶𝐷 is a rectangle.