Final answer:
It is true that a parallelogram with diagonals that are equal and perpendicular bisectors of each other is a square. The diagonals indicate equal-length sides and right angles, fulfilling the criteria for a square. Vector mathematics also corroborates these geometric properties through vector addition and resultant vector calculations.
Step-by-step explanation:
The question asks whether a parallelogram with equal diagonals that are also perpendicular bisectors of each other is necessarily a square. The statement is True. If the diagonals of a parallelogram are equal, it means that they bisect each other into equal halves, indicating that all four sides of the parallelogram are of equal length. Moreover, if these diagonals are perpendicular bisectors, it means each angle of the parallelogram is a right angle. Therefore, having four right angles and four sides of equal length, such a figure meets all requirements to be a square.
To apply these concepts with vectors:
- Using the Pythagorean theorem is True for calculating the length of the resultant vector when two vectors are at right angles to each other.
- Vectors can indeed form the shape of a right-angle triangle with their x and y components, which is True.
- For two vectors that are perpendicular to each other, they form a 90° angle (True).
- When comparing vector relationships, such as in the understanding of parallelograms in vector addition, it is possible to find scenarios where vectors are mutually perpendicular to each other (True).
- Furthermore, the concept of similar triangles and the properties related to proportional sides can be used in vector mathematics.
- The solution provided in the question shows practical application of vector addition, using measurement tools to determine the length and direction of resultant vectors.
- Lastly, knowing only the angles of vectors, one can determine the angle of the resultant vector through vector addition (True).