We need to find the legs of a right triangle.
We know that one of the acute angles is 45º. Since the three internal angles of any triangle must add up to 180º, the other angle A is given by:
[tex]\begin{gathered} A+45\degree+90\degree=180\degree \\ \\ A+135\degree=180\degree \\ \\ A+135\degree-135\degree=180\degree-135\degree \\ \\ A=45\degree \end{gathered}[/tex]
So, both acute angles measure 45º. Since those angles are congruent, so are the legs of the triangle.
Now, calling x the length of each leg, we can use the Pythagorean Theorem to find:
[tex]\begin{gathered} (leg_1)^2+(leg_2)^2=(\text{ hypotenuse})^{2} \\ \\ x^{2}+x^{2}=4^{2} \\ \\ 2x^{2}=4^{2} \\ \\ \frac{2x^2}{2}=\frac{4^2}{2} \\ \\ x^{2}=\frac{4^2}{2} \\ \\ \sqrt[]{x^{2}}=\sqrt[]{\frac{4^2}{2}} \\ \\ x=\frac{4}{\sqrt[]{2}} \end{gathered}[/tex]
Therefore, the lengths of the legs are:
[tex]\frac{4}{\sqrt[]{2}}[/tex]
