Question

what is the approximate side length of the square?
3.0 units
3.5 units
4.2 units
4.9 units

Explanation:

Step1: Recall diagonal - side relationship

In a square, if the side length is $s$ and the length of the diagonal is $d$, then $d^{2}=2s^{2}$ (by Pythagorean theorem, since for a square with side $s$, $s^{2}+s^{2}=d^{2}$). Here, the length of the half - diagonal is 3, so the length of the full diagonal $d = 6$.

Step2: Solve for side length

We have $d^{2}=2s^{2}$, substituting $d = 6$, we get $6^{2}=2s^{2}$, which simplifies to $36 = 2s^{2}$. Then $s^{2}=\frac{36}{2}=18$. So, $s=\sqrt{18}\approx4.24$.

Answer:

4.2 units