Two rectangles have the same area and the same perimeter. Does this mean that they are congruent to each other?
Yes, they are congruent.
Explanation
If two rectangles have the same perimeter, the sum of their length and breadth is equal. If they have the same area, the product of their length and breadth is also equal. Since the sum and product of the dimensions of both rectangles are identical, the dimensions themselves must be the same. Therefore, the rectangles have the exact same shape and size, making them congruent.
Solution Steps
Step 1: Let the dimensions of the two rectangles be l, b and x, y.
Step 2: Since perimeters are equal, 2(l + b) = 2(x + y), which means l + b = x + y.
Step 3: Since areas are equal, l × b = x × y.
Step 4: Since the sum and product of the dimensions are equal, the sets of dimensions {l, b} and {x, y} must be identical. Thus, the rectangles are congruent.