Question

1. solve for a and b.

a = ____
b = ____

1. solve for w and 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 $a:b:c = 1:1:\sqrt{2}$, where $a$ and $b$ are the legs and $c$ is the hypotenuse.

Step2: Set up the relationship for the hypotenuse

We know that $c=\sqrt{14}$ and $c = \sqrt{2}a$ (since $a = b$ in a 45 - 45-90 triangle). So, $\sqrt{14}=\sqrt{2}a$.

Step3: Solve for $a$ and $b$

To solve for $a$, we divide both sides of the equation $\sqrt{14}=\sqrt{2}a$ by $\sqrt{2}$: $a=\frac{\sqrt{14}}{\sqrt{2}}=\sqrt{\frac{14}{2}}=\sqrt{7}$. Since $a = b$ in a 45 - 45-90 triangle, $b=\sqrt{7}$.

Answer:

$a=\sqrt{7}$
$b=\sqrt{7}$