Question

question
find the length of side x in simplest radical form with a rational denominator.
triangle image: right triangle with two 45° angles, hypotenuse 2, leg x

Explanation:

Step1: Identify triangle type

This is a 45 - 45 - 90 right triangle (two angles are $45^{\circ}$, one is $90^{\circ}$), so it's an isosceles right triangle. In a 45 - 45 - 90 triangle, the legs are equal, and the hypotenuse $h$ is related to a leg $l$ by $h = l\sqrt{2}$. Here, the hypotenuse is 2, and we need to find a leg $x$.

Step2: Set up the formula

Let the leg length be $x$. The hypotenuse formula for 45 - 45 - 90 triangle is $h=x\sqrt{2}$, where $h = 2$. So we have $2=x\sqrt{2}$.

Step3: Solve for $x$

To solve for $x$, we divide both sides by $\sqrt{2}$: $x=\frac{2}{\sqrt{2}}$. Then rationalize the denominator by multiplying numerator and denominator by $\sqrt{2}$: $x=\frac{2\sqrt{2}}{\sqrt{2}\times\sqrt{2}}=\frac{2\sqrt{2}}{2}=\sqrt{2}$.

Answer:

$\sqrt{2}$