Final answer:
To prove that the parallelogram is a rhombus, we consider the properties of vectors and the Pythagorean theorem. The side length can be determined using the diagonals and applying the theorem, which relates the lengths of the legs and the hypotenuse of a right triangle.
Step-by-step explanation:
To show that a parallelogram is a rhombus and determine its side length, we need to use the properties of vectors and the Pythagorean theorem. In the given scenario, the parallelogram is constructed using vectors A and B, with diagonals A + B and A - B. If all sides of the parallelogram are equal, then it is a rhombus. According to the information provided, the longer diagonal D has been measured to be 16.2 cm (difference of vectors), while the resultant R (sum of vectors) is 5.8 cm.
Using the Pythagorean theorem, which states that in a right triangle the sum of the squares of the two legs (a and b) is equal to the square of the hypotenuse (c), we can calculate the side length of the rhombus. The theorem is mathematically represented as a² + b² = c² and can be rewritten to solve for c: c = √(a² + b²).
To find the side length s of the rhombus, we would equate the diagonal measurements to a right triangle scenario where s would be the hypotenuse and the diagonals would serve as the legs. Thus, the side length s can be given by s = √((D/2)² + (R/2)²), since each diagonal bisects the rhombus into two congruent right triangles