The perimeter of a square is 56 cm. What is the approximate length of its diagonal?
The problem provides the perimeter of a square as \(P = 56\text{ cm}\). We know that a square has four equal sides, and its perimeter is given by the formula \(P = 4s\), where \(s\) is the side length. To find the side length \(s\), we divide the total perimeter by 4:
\[s = \frac{56\text{ cm}}{4} = 14\text{ cm}\]
The diagonal \(d\) of a square can be found using the Pythagorean theorem, relating the side length \(s\) and the diagonal:
\[d^2 = s^2 + s^2 = 2s^2\]
Taking the square root of both sides gives the formula for the diagonal:
\[d = s\sqrt{2}\]
Substituting the value \(s = 14\text{ cm}\) into this formula, we get:
\[d = 14\sqrt{2}\text{ cm}\]
To find the approximate numerical value, we use the approximation \(\sqrt{2} \approx 1.414\):
\[d \approx 14 \times 1.414 = 19.796\text{ cm}\]
Rounding this value, we find that the diagonal is approximately \(19.8\text{ cm}\). Comparing this result with the given options (a. 19.8 cm, b. 10.6 cm, c. 14.0 cm), we identify that the correct answer is (a).
Understanding how to calculate the diagonal of a square is highly practical in home improvement and construction. For example, if you are building a square table or tiling a floor and need to check that your corners are perfectly square, you can measure the diagonals. If the measured diagonals of your square frame are equal to \(s\sqrt{2}\), where \(s\) is the side length, then you know your frame is geometrically perfect!