Final answer:
Upon computing the lengths of sides PR and QS of parallelogram PQRS using the distance formula, it is found that PR \(\\eq\) QS, indicating PQRS is not a rhombus because not all sides are equal.
Step-by-step explanation:
To prove that parallelogram PQRS with vertices Q(8, 5), R(5, 1), and S(2, 5) is a rhombus, we need to demonstrate that all four sides are of equal length. In a parallelogram, opposite sides are parallel and equal in length by definition. The next step is to compute the distances between the points Q, R, and S to find PR and QS.
Using the distance formula \(d = \sqrt{(x_2-x_1)^2+(y_2-y_1)^2}\), the length of PR is:
\(d_{PR} = \sqrt{(5-2)^2+(1-5)^2} = \sqrt{3^2+(-4)^2} = \sqrt{9+16} = \sqrt{25} = 5\)
The length of QS can be found similarly:
\(d_{QS} = \sqrt{(8-2)^2+(5-5)^2} = \sqrt{6^2+0^2} = \sqrt{36} = 6\)
Since PR \(\\eq\) QS, based on different lengths, the parallelogram PQRS is not a rhombus because the sides PR and QS are not equal. In a rhombus, all sides must have equal length. Therefore, without the need for further proof, we can say that PQRS is not a rhombus due to the unequal lengths of its sides.