Final answer:
The statement that a parallelogram with congruent diagonals is always a rectangle is false, because while a rectangle does have congruent diagonals, not all parallelograms with congruent diagonals are rectangles; they could also be rhombi. Option B is the correct answer.
Step-by-step explanation:
If the diagonals of a parallelogram are congruent, then the statement that it must also be a rectangle is false. The defining property of a rectangle is that all angles are right angles (90 degrees), not necessarily that the diagonals are congruent. However, in a rectangle, it just so happens that the diagonals are congruent, but the converse is not always true. A parallelogram with congruent diagonals could be a rectangle, but it could also be another type of parallelogram called a rhombus, which also has congruent diagonals. What distinguishes a rectangle from a rhombus is the internal angles, not the congruence of the diagonals.
To clarify with examples, let's use the provided statements which relate to vectors and geometric shapes:
- We can use the Pythagorean theorem to calculate the length of the resultant vector obtained from the addition of two vectors which are at right angles to each other. This concept is used in forming a right-angle triangle from vector components and is indeed true.
- A vector can form the shape of a right-angle triangle with its x and y components, which is also true. This corresponds to the vector's horizontal and vertical components.
- The displacement of a person walking is independent of the path taken, as displacement is a vector quantity that depends solely on the starting and ending points. Therefore, two paths that start and end at the same points have equal displacement, making such a statement false.
Back to the original statement, while the properties of vectors and right-angle triangles can easily be determined as true or false, the statement about the parallelogram is more nuanced.