Consider a right-angled triangle, where one of the acute angles measures 45º. If the hypotenuse measures 10cm, what is the length of the legs?

Let the vertical leg be b; The vertical leg is given by; $$\cos{45^\circ}=\frac{b}{10}$$ Solve for b; $$b=5\sqrt{2}$$ now let the horizontal leg be a. Using the Pythagorean theorem, the length of the horizontal leg is; $$a^2+(5\sqrt2)^2=10^2$$ Solve for a; $$a=5\sqrt2$$ So the length of the legs is $$5\sqrt2$$