I need help checking my answers.
Which of the following is a true statement?
A. It is possible for two rectangles to have the same area, but only if they have the same perimeter.
B. It is possible for two rectangles to have the same area without having the same perimeter.
C. It is possible for two squares to have the same area without having the same perimeter.***
D. It is possible for two squares to have the same perimeter without having the same area.
Your choice of C is wrong!
If two squares have the same area, then the two squares are congruent,
which means they must have the same sides, thus the same perimeter.
why don't you make sketches of situations for each case ?
Alright, thanks for checking. ^w^
To check for the correct answer, we need to assess each statement individually. Let's analyze each option:
A. It is possible for two rectangles to have the same area, but only if they have the same perimeter.
This statement suggests that the two rectangles must have the same perimeter to have the same area. To check if this is true, we can visualize different scenarios where two rectangles have the same area but different perimeters. For example, consider a rectangle with dimensions 2x4 and another with dimensions 1x8. These rectangles both have an area of 8 but different perimeters (12 and 18, respectively). Therefore, option A is not correct.
B. It is possible for two rectangles to have the same area without having the same perimeter.
This statement proposes that two rectangles can have the same area even if their perimeters differ. To verify the accuracy of this statement, we can examine various scenarios where rectangles have the same area and different perimeters. For instance, consider a rectangle with dimensions 3x4 and another with dimensions 2x6. These rectangles both have an area of 12 but distinct perimeters (14 and 16, respectively). Thus, option B is correct.
C. It is possible for two squares to have the same area without having the same perimeter.
This statement addresses squares specifically, suggesting that two squares can have the same area while having different perimeters. We can easily validate this by considering two squares with different side lengths. For example, take a square with side length 3 and another with side length 6. Although they have different perimeters (12 and 24, respectively), both squares have an area of 9. Thus, option C is correct.
D. It is possible for two squares to have the same perimeter without having the same area.
This statement states that two squares can have the same perimeter but different areas. To verify this, we can consider squares with equal perimeters but varying side lengths. For instance, if we take a square with side length 2 (perimeter 8) and another with side length 4 (perimeter 16), we can see that their areas differ (4 and 16, respectively). Hence, option D is correct.
Based on the above analysis, options B, C, and D are all true statements. Therefore, the correct answer is option C: It is possible for two squares to have the same area without having the same perimeter.