The area of a rectangle is 100 square inches, and the perimeter is 40 inches. A second rectangle has the same area but a different perimeter. Is the second rectangle a square? Explain why or why not.

Area = 100Sq/in Perimeter = 40
Easiest side length would be 10in x 10in = 100sq/in (ie a square)
Instead of a square, the 100sq in could be 100 laid out in a long row: the perimeter therefore would be 100 + 1 + 100 + 1 = 202in. Not a square

Answered by Trim | 2024-06-10

Answer:
So, the second rectangle is not necessarily a square.
Step-by-step explanation:
No, the second rectangle is not necessarily a square.
For the first rectangle, we can find the length and width by solving the system of equations: length x width = 100 2(length + width) = 40
This simplifies to: length + width = 20
Using substitution, we can solve for one of the variables: length = 20 - width
Substituting into the first equation: (20 - width) x width = 100
Solving for width: width^2 - 20width + 100 = 0 (width - 10)(width - 10) = 0 width = 10, length = 10
So, the first rectangle is a square with a side length of 10 inches.
Now, for the second rectangle, we know it has the same area as the first rectangle, which is 100 square inches. Let's call its length L and width W.
We also know that its perimeter is different from the first rectangle. Let's call it perimeter P.
So, we have two equations: L x W = 100 2(L + W) ≠ 40
We can simplify the second equation to: L + W ≠ 20
Now, we can find an example of a rectangle with the same area but a different perimeter. For instance, we can choose a rectangle with a length of 20 inches and a width of 5 inches. This rectangle has an area of 100 square inches, but its perimeter is: 2(20 + 5) = 50 inches
The second rectangle does not necessarily have equal measurements on all sides. Hence, it cannot be classified as a square.

Answered by noeckahenry | 2024-06-24

The second rectangle with the same area as the first (100 square inches) is not necessarily a square because rectangles can have identical areas while varying in dimensions. The perimeter of a rectangle changes based on side lengths, thus a square is just one specific case of a rectangle. Other configurations, such as elongated rectangles, would have different perimeters while still achieving the same area.
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Answered by Trim | 2024-12-26