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A parallelogram has vertices at the coordinates 𝐴 (−4, −1), 𝐵 (0, −3), 𝐶 (−1, −5), and 𝐷 (−5, −3). Work out the length of the diagonal 𝐴𝐶. Work out the length of the diagonal 𝐵𝐷. Using these lengths, is the parallelogram 𝐴𝐵𝐶𝐷 a rectangle?
A parallelogram has vertices at the coordinates 𝐴 negative four, negative one; 𝐵 zero, negative three; 𝐶 negative one, negative five; and 𝐷 negative five, negative three. Work out the length of the diagonal 𝐴𝐶. Work out the length of the diagonal 𝐵𝐷. Using these lengths, is the parallelogram 𝐴𝐵𝐶𝐷 a rectangle?
So we’ve been given the coordinates of the four vertices of a parallelogram and asked to find the length of its two diagonals. To do this, we’ll need to recall the distance formula, which tells us how to calculate the distance between two points on a coordinate grid.
If the two points have coordinates 𝑥 one 𝑦 one and 𝑥 two 𝑦 two, then the distance between them 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 sides of a right-angled triangle and 𝑑 is the hypotenuse.
To find the length of the diagonal 𝐴𝐶, we need to substitute the coordinates for 𝐴 and 𝐶 into the distance formula. Now there are also a lots of negatives involved here, so we need to be careful with the signs. We have that 𝐴𝐶 is equal to the square root of negative one minus negative four squared plus negative five minus negative one squared.
This is equal to the square root of three squared plus negative four squared. Three squared is nine and negative four squared is 16. So we have the square root of nine plus 16, which is equal to the square root of 25. 25 is a square number and its square root is exactly equal to five. So we’ve found the length of the first diagonal 𝐴𝐶, and now we need to find the length of the second diagonal 𝐵𝐷.
We we’ll substitute the coordinates for 𝐵 and 𝐷 into the distance formula. Again, we need to be very careful with the negative signs. We have that 𝐵𝐷 is equal to the square root of negative five minus zero squared plus negative three minus negative three squared. This simplifies to the square root of negative five squared plus zero squared. Negative five squared is 25 and zero squared is zero, so we have the square root of 25 which is equal to five.
You will have noticed, I’m sure, that the length of the two diagonals of this parallelogram are the same. They’re both equal to five units. How does this help us with answering the final part of the question? Well, a key fact which is true of rectangles but isn’t true of parallelograms in general is that the diagonals are equal in length.
We’ve already calculated that 𝐴𝐶 and 𝐵𝐷 are the same length. They are both equal to five and, therefore, this tells us that the parallelogram 𝐴𝐵𝐶𝐷 is a rectangle.
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