Question

The figure below is a square. Find the length of side x in simplest radical form with a rational denominator.

Answer 1

Answer:

$\frac{\sqrt{10}}{2}$

Step-by-step explanation:

The diagonal forms two 45-45-90 triangles, with the diagonal being the hypotenuse of both. The Pythagorean Theorem states that $a^2+b^2=c^2$, where $c$ is the hypotenuse of the triangle, and $a$ and $b$ are the two legs of the triangle.

From the Isosceles Base Theorem, the two legs of a 45-45-90 triangle are always equal. Since we're given a diagonal of $\sqrt{5}$, we have:

$x^2+x^2=\sqrt{5}^2,\\2x^2=5,\\x^2=\frac{5}{2},\\x=\sqrt{\frac{5}{2}}=\frac{\sqrt{5}}{\sqrt{2}}=\frac{\sqrt{5}}{\sqrt{2}}\cdot \frac{\sqrt{2}}{\sqrt{2}}=\boxed{\frac{\sqrt{10}}{2}}$