Question

solve for y.

Explanation:

Step1: Identify triangle type

It's a 45 - 45 - 90 right - triangle. In a 45 - 45 - 90 triangle, the ratio of the sides is $1:1:\sqrt{2}$, where the hypotenuse $c$ is related to the legs $a = b$ by $c=a\sqrt{2}$.

Step2: Set up the equation

Given the hypotenuse $c = 10$ and $c=y\sqrt{2}$ (since the legs of a 45 - 45 - 90 triangle are equal and we want to find $y$ which is a leg). So, $10=y\sqrt{2}$.

Step3: Solve for $y$

Divide both sides of the equation $10=y\sqrt{2}$ by $\sqrt{2}$: $y=\frac{10}{\sqrt{2}}$. Rationalize the denominator by multiplying the numerator and denominator by $\sqrt{2}$: $y=\frac{10\sqrt{2}}{2}=5\sqrt{2}$.

Answer:

$5\sqrt{2}$