Question

the perimeter of a square is 56 cm. what is the approximate length of its diagonal? 10.6 cm 14.0 cm 15.0 cm 19.8 cm

Explanation:

Step1: Find the side length of the square

The perimeter formula of a square is \(P = 4s\) (where \(P\) is the perimeter and \(s\) is the side - length). Given \(P=56\) cm, then \(s=\frac{P}{4}\).
\(s=\frac{56}{4}=14\) cm.

Step2: Use the Pythagorean theorem to find the diagonal length

In a square, if the side - length is \(s\) and the diagonal length is \(d\), by the Pythagorean theorem \(d^{2}=s^{2}+s^{2}\) (since in a square \(a = b=s\) in \(a^{2}+b^{2}=c^{2}\), and \(c = d\)). So \(d^{2}=2s^{2}\).
Substitute \(s = 14\) into the formula: \(d^{2}=2\times14^{2}=2\times196 = 392\). Then \(d=\sqrt{392}\).
We know that \(\sqrt{392}=\sqrt{49\times8}=7\sqrt{8}=7\times2\sqrt{2}=14\sqrt{2}\approx14\times1.414 = 19.796\approx19.8\) cm.

Answer:

\(19.8\) cm (the fourth option)