Question

answers in exact form.
what patterns do you notice?

Explanation:

Step1: Identify the triangle type

This is a 45 - 45 - 90 right - triangle. In a 45 - 45 - 90 triangle, the ratio of the sides is $1:1:\sqrt{2}$, where the legs are of equal length and the hypotenuse $c$ is related to a leg $a$ by $c = a\sqrt{2}$.

Step2: Find the value of $x$ and $y$

Let the length of each leg be $x = y$. Given the hypotenuse $c=4\sqrt{2}$. Using the formula $c = a\sqrt{2}$, we set $a\sqrt{2}=4\sqrt{2}$. Solving for $a$ (where $a = x=y$), we divide both sides of the equation $a\sqrt{2}=4\sqrt{2}$ by $\sqrt{2}$. So $a=\frac{4\sqrt{2}}{\sqrt{2}} = 4$.

Answer:

$x = 4$
$y = 4$