Sure, let's work through this problem step by step to find the approximate length of the diagonal of the square with a given perimeter of 56 cm.
### Step-by-Step Solution:
1. Understanding the Perimeter:
- The perimeter of a square is the total length around the square.
- For a square with side length [tex]\( s \)[/tex], the perimeter [tex]\( P \)[/tex] is given by the formula:
[tex]\[
P = 4s
\][/tex]
2. Calculating the Side Length:
- Given the perimeter [tex]\( P = 56 \)[/tex] cm, we can find the side length [tex]\( s \)[/tex] of the square using the formula:
[tex]\[
s = \frac{P}{4}
\][/tex]
- Substituting the given perimeter:
[tex]\[
s = \frac{56}{4} = 14 \text{ cm}
\][/tex]
3. Finding the Diagonal:
- The diagonal [tex]\( d \)[/tex] of a square can be found using the Pythagorean theorem. For a square with side length [tex]\( s \)[/tex], the diagonal forms a right triangle with the two sides of the square. The length of the diagonal [tex]\( d \)[/tex] is given by:
[tex]\[
d = s\sqrt{2}
\][/tex]
- Substituting [tex]\( s = 14 \)[/tex] cm:
[tex]\[
d = 14 \times \sqrt{2}
\][/tex]
- Approximating [tex]\( \sqrt{2} \approx 1.414 \)[/tex], we get:
[tex]\[
d \approx 14 \times 1.414 = 19.8 \text{ cm}
\][/tex]
So, the approximate length of the diagonal of the square is 19.8 cm.
### Checking the Options:
- 106 cm: This is too large.
- 140 cm: This is also too large.
- 15.0 cm: This is too small.
- 198 cm: This is much too large and seems to be a misinterpretation of the diagonal length.
Hence, none of the provided options (106 cm, 140 cm, 15.0 cm, 198 cm) correctly match the approximate diagonal length of 19.8 cm. If this value were in the options, it would be the correct answer.
