if you fold a square paper vertically, the new rectangle has a perimeter of 39cm. what is the area of the original square?
what is the perimeter of the original square?
what is the area of the resulting rectangle?
make a ratio of area and perimeter. what do you notice?
To solve this problem step-by-step:
Step 1: Let's assume the side length of the original square is "x" cm.
Step 2: If we fold the square paper vertically, the resulting rectangle will have one dimension equal to "x" cm, and the other dimension equal to half of "x" cm.
Step 3: Given that the perimeter of the resulting rectangle is 39 cm, we can set up the equation:
2(x/2) + x = 39
Simplifying the equation, we have:
x + x = 39
2x = 39
x = 39/2
x = 19.5 cm
Step 4: The perimeter of the original square is equal to the sum of all four sides, so it is:
4x = 4 * 19.5 = 78 cm
Step 5: The area of the original square is given by the formula:
Area = side length * side length
Area = 19.5 * 19.5 = 380.25 cm²
Step 6: The area of the resulting rectangle is equal to the product of its dimensions:
Area = x * (x/2)
Area = 19.5 * 9.75 = 189.375 cm²
Step 7: The ratio of the area to the perimeter is given by:
Ratio = Area / Perimeter
Ratio = 380.25 / 78 = 4.878205128205128
When comparing the area and perimeter ratios, we notice that the area ratio is greater than the perimeter ratio.
To solve these questions, we can follow a step-by-step approach:
1. Let's assume the side length of the original square is "s" cm.
2. When we fold the square paper vertically, the resulting rectangle will have a length equal to "s" cm and a width equal to half of "s" cm, which is (1/2)s cm.
3. Given that the perimeter of the resulting rectangle is 39 cm, we can use the formula for the perimeter of a rectangle: 2(length + width).
Therefore, 2(s + (1/2)s) = 39 cm.
Solving this equation, we get (5/2)s = 39 cm.
Dividing both sides by (5/2), we find s = 39 x (2/5) = 15.6 cm.
So, the side length of the original square is 15.6 cm.
4. The perimeter of a square is calculated by multiplying the side length by 4. Therefore, the perimeter of the original square is 4 x 15.6 cm = 62.4 cm.
5. The area of a square is calculated by multiplying the side length by itself. Thus, the area of the original square is 15.6 cm x 15.6 cm = 243.36 cm².
6. The area of the resulting rectangle is found by multiplying its length and width: 15.6 cm x (1/2) x 15.6 cm = 121.44 cm².
7. To find the ratios, we divide the area by the perimeter for both the original square and the resulting rectangle.
For the square: 243.36 cm² / 62.4 cm = 3.9 cm²/cm.
For the rectangle: 121.44 cm² / 39 cm = 3.12 cm²/cm.
From the ratios, we notice that the original square has a higher area-to-perimeter ratio (3.9 cm²/cm) compared to the resulting rectangle (3.12 cm²/cm).
area- 169
perimeter 52
area of rectangle- 84.5
169:52= 3.25
84.5: 39= 2.166666666