### Video Transcript

A parallelogram π΄π΅πΆπ· has vertices π΄: negative five, five; π΅: nine, three; πΆ: eight, negative four; and π·: negative six, negative two. Firstly, calculate the length of π΄πΆ. Give an exact answer. Secondly, calculate the length of π΅π·. Give an exact answer. Hence, state whether or not the parallelogram is a rectangle.

So weβve been given the coordinates of the four vertices of a parallelogram and asked to calculate the lengths of π΄πΆ and π΅π·, which are the diagonals of the parallelogram. To answer this question, weβll need to use the distance formula, which tells us how to find the distance between two points on a coordinate grid. Letβs recall its definition.

The distance π between two points with coordinates π₯ one, π¦ one and π₯ two, π¦ two is given by the square root of π₯ two minus π₯ one squared plus π¦ two minus π¦ one squared. This is an application of the Pythagorean theorem, where π₯ two minus π₯ one and π¦ two minus π¦ one are the horizontal and vertical lengths of a right-angled triangle and π is the length of the third side, the hypotenuse.

To find the length of π΄πΆ, first of all, we need to substitute the coordinates of π΄ and πΆ into the distance formula. We have then that π 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 π΄πΆ. So weβll leave it in terms of a surd, the square root of 250.

Next, letβs calculate the length of π΅π· by substituting the coordinates of the points π΅ and π· into the distance formula. We have that π 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 π΄πΆ and π΅π·. And you will have noticed, Iβm sure, that they are in fact the same. Theyβre both equal to the square root of 250. How does this help us with answering the final part of the question? Well, remember, π΄πΆ and π΅π· are the diagonals of this parallelogram. A key fact about rectangles, which isnβt true of parallelograms in general, is that their diagonals are equal. Therefore, as the diagonals of this parallelogram are equal in length, π΄π΅πΆπ· is a rectangle.