. Two groups of students are using a dynamic geometry software program to investigate the properties of quadrilaterals. The first group has concluded that the diagonals of a quadrilateral always bisect each other. The second group of students is not convinced. Which of the following should the teacher encourage the students to do to resolve the apparent contradiction and Why? A-D are answer choice
A. Assume that the diagonals of a quadrilateral don't bisect each other and derive a contradiction.
B. Find an example of a quadrilateral with diagonals that do not bisect each other.
C. Formally prove that if the diagonals of a quadrilateral bisect each other, the quadrilateral is a parallelogram.
D. Use informal reasoning to show that the diagonals of a quadrilateral create
Personally, I would do B.
C is theoretically attractive but it took me about five seconds to do B so I would say B.
To resolve the apparent contradiction, the teacher should encourage the students to choose option C: Formally prove that if the diagonals of a quadrilateral bisect each other, the quadrilateral is a parallelogram.
Here's why:
Option A suggests assuming the diagonals of a quadrilateral don't bisect each other and deriving a contradiction. While this approach can be useful in some cases, it might not convince the second group of students since it relies on assumptions rather than concrete evidence.
Option B suggests finding an example of a quadrilateral with diagonals that do not bisect each other. While this can help provide counterexamples, it doesn't give a broader understanding of the relationship between diagonals and quadrilaterals in general. The second group of students might argue that this is just an exception, and it doesn't disprove the statement for all quadrilaterals.
Option C, on the other hand, suggests formally proving that if the diagonals of a quadrilateral bisect each other, the quadrilateral is a parallelogram. This approach requires clear and logical reasoning, providing a solid foundation for understanding the relationship between diagonals and quadrilaterals.
By proving this statement, the second group of students will gain a deeper understanding of the concept and be able to see the logical connection between the diagonals bisecting and the quadrilateral being a parallelogram. This will help them resolve their doubt and reconcile the apparent contradiction.