A right triangle has a hypotenuse of length 35. The ratio of the length of the shorter leg to the longer leg of the triangle is 3:4. What is the length of the shorter leg of the triangle?

As given, 3:4 is the ratio of shorter leg to longer leg. We assume that hypotenuse is 5. It always works that way: 3:4:5, 5 as the hypotenuse. However this is only in right triangles. Ok so 5 is 35. What we need is the 3. Set up a proportion as such:
(3/5)=(x/35). Cross multiply so you get 3*35 which is 105. Divide it by 5. 105/5=21. 21 is your answer.

\( x-unit\ of\ length\\\\3x;\ 4x-legs\\\\Use\ Pythagoras\ Theorem:\\\\(3x)^2+(4x)^2=35^2\\9x^2+16x^2=1225\\25x^2=1225\\x^2=1225:25\\x^2=49\\x=\sqrt{49}\\x=7\\\\Answer: Shorter\ leg\ has\ length\ 3\cdot7=21. \)