Question

solve for k.
answer attempt 1 out of 3
k =

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 $1:1:\sqrt{2}$. Let the legs of the right - triangle be of length $a$ (in this case, one leg is $k$ and the other is the side adjacent to the 45 - degree angle), and the hypotenuse be $c$. The relationship is $c = a\sqrt{2}$.

Step2: Set up the equation

We know that the hypotenuse $c = 12$, and $c=k\sqrt{2}$ (since the legs of a 45 - 45-90 triangle are equal, and if one leg is $k$, using the Pythagorean theorem $a^{2}+a^{2}=c^{2}$, or $2a^{2}=c^{2}$, and $c = a\sqrt{2}$). So, $12 = k\sqrt{2}$.

Step3: Solve for $k$

To solve for $k$, we divide both sides of the equation $12 = k\sqrt{2}$ by $\sqrt{2}$. We get $k=\frac{12}{\sqrt{2}}$. Rationalize the denominator by multiplying the numerator and denominator by $\sqrt{2}$: $k=\frac{12\sqrt{2}}{2}=6\sqrt{2}$.

Answer:

$6\sqrt{2}$