Question

solve for x and y

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 $\sqrt{2}$ times the length of a leg $a$ (i.e., $c = a\sqrt{2}$).

Step2: Set up the equation for the hypotenuse

Let the length of each leg be $x = y$. The hypotenuse $c=11\sqrt{7}$. Since $c = a\sqrt{2}$, we have $11\sqrt{7}=x\sqrt{2}$.

Step3: Solve for $x$ and $y$

To find $x$ (and since $y = x$), we can solve the equation $11\sqrt{7}=x\sqrt{2}$ for $x$. Cross - multiply to get $x=\frac{11\sqrt{7}}{\sqrt{2}}$. Rationalize the denominator: $x=\frac{11\sqrt{7}\times\sqrt{2}}{\sqrt{2}\times\sqrt{2}}=\frac{11\sqrt{14}}{2}$. So $y=\frac{11\sqrt{14}}{2}$ as well.

Answer:

$x=\frac{11\sqrt{14}}{2}$
$y=\frac{11\sqrt{14}}{2}$