Use the relationships in 45-45-90 triangles to solve the following problem. If the hypotenuse of a triangle is 4 cm, what is the lenght of the leg?

In a 45-45-90 triangle, the legs are of equal length, and the relationship between the legs (let's call each leg \( x \)) and the hypotenuse \( h \) is given by the formula:

\[
h = x\sqrt{2}
\]

Given that the hypotenuse \( h = 4 \) cm, we can set up the equation:

\[
4 = x\sqrt{2}
\]

To solve for \( x \), divide both sides by \( \sqrt{2} \):

\[
x = \frac{4}{\sqrt{2}}
\]

Now, to simplify \( \frac{4}{\sqrt{2}} \), we can multiply the numerator and the denominator by \( \sqrt{2} \):

\[
x = \frac{4\sqrt{2}}{2} = 2\sqrt{2}
\]

Thus, the length of each leg of the triangle is:

\[
\boxed{2\sqrt{2}} \text{ cm}
\]

u sure

Yes, I am sure! Let's review the solution step-by-step to ensure accuracy:

For a 45-45-90 triangle, the relationship between the length of the legs \(x\) and the hypotenuse \(h\) is expressed as:

\[
h = x\sqrt{2}
\]

Given that the hypotenuse \(h\) is 4 cm, we can substitute this value into the equation:

\[
4 = x\sqrt{2}
\]

To find the length of the leg \(x\), we can rearrange the equation:

\[
x = \frac{4}{\sqrt{2}}
\]

Next, we simplify \( \frac{4}{\sqrt{2}} \). To do this, we multiply the numerator and the denominator by \( \sqrt{2} \) to rationalize the denominator:

\[
x = \frac{4 \sqrt{2}}{2} = 2\sqrt{2}
\]

So, the length of each leg of the triangle is indeed:

\[
\boxed{2\sqrt{2}} \text{ cm}
\]

Feel free to ask if you have any further questions!