Math Snap

STEP 1

1. The triangle is a 45-45-90 triangle, meaning it is an isosceles right triangle.
2. In a 45-45-90 triangle, the legs are equal in length, and the hypotenuse is $x\sqrt{2}$, where $x$ is the length of each leg.

STEP 2

1. Understand the properties of a 45-45-90 triangle.
2. Set up an equation based on the relationship between the legs and the hypotenuse.
3. Solve for $x$.

STEP 3

Recognize that in a 45-45-90 triangle, the legs are equal, and the hypotenuse is $x\sqrt{2}$, where $x$ is the length of each leg.

STEP 4

Set up the equation using the relationship of the hypotenuse to the legs in a 45-45-90 triangle. Given that the hypotenuse is $\sqrt{6}$, we have:
$$x\sqrt{2} = \sqrt{6}$$

STEP 5

Solve for $x$ by dividing both sides of the equation by $\sqrt{2}$:
$$x = \frac{\sqrt{6}}{\sqrt{2}}$$

STEP 6

Simplify the expression by rationalizing the denominator:
$$x = \frac{\sqrt{6}}{\sqrt{2}} \times \frac{\sqrt{2}}{\sqrt{2}}$$ $$= \frac{\sqrt{12}}{2}$$

Simplify $\sqrt{12}$ to find the final expression for $x$:
$$\sqrt{12} = \sqrt{4 \times 3} = \sqrt{4} \times \sqrt{3} = 2\sqrt{3}$$ Thus,
$$x = \frac{2\sqrt{3}}{2} = \sqrt{3}$$ The measure of $x$ is:
$$x = \sqrt{3}$$