Question

If you are given two rectangles with the same perimeter, are the two shapes congruent?

In your answer, include the following:
•State whether or not you think the two shapes are congruent.
•Give an example to support your answer.
•Explain why your example support your statement.

Answer 1

Two rectangles with the same perimeter are not necessarily congruent because congruence requires identical shapes and sizes, whereas identical perimeters do not ensure identical dimensions.

Step-by-step explanation:

Two rectangles with the same perimeter are not necessarily congruent because congruence implies that all corresponding sides and angles are identical. To be congruent, two shapes must be the same shape and size. However, only sharing the same perimeter does not guarantee that all dimensions are identical. For example, a rectangle that is 2 units by 6 units and another that is 3 units by 5 units both have a perimeter of 16 units, yet they are not congruent because their dimensions differ.

This concept relates to the mathematical understanding that while perimeter deals with the total boundary length, congruency is about exact shape and size. Therefore, rectangles can have identical perimeters but different dimensions, leading to different shapes and areas, and hence, they are not congruent.