The larger leg of a right triangle is 3cm longer than its smaller leg. The hypotenuse is 6cm longer than the smaller leg. how many centimeters long is the smaller leg   please explain how you got it 

[latex]b=a+3\\ c=a+6\\ a^2+b^2=c^2\\\\ a^2+(a+3)^2=(a+6)^2\\ a^2+a^2+6a+9=a^2+12a+36\\ a^2-6a-27=0\\ a^2-9a+3a-27=0\\ a(a-9)+3(a-9)=0\\ (a+3)(a-9)=0\\ a=-3 \vee a=9\\ [/latex] The length can't be negative, so the answer is 9 cm.

Let the smaller leg be x. Thus, larger leg = 3 +x Thus, hypotenuse = 6 + x Applying pythagoras theorum ; [latex](6+x)^2 = x^2 + (3+x)^2 \\ \\ or, 6^2 + 12x + x^2 = 3^2 + 6x + x^2 \\ \\ i.e., 36 + 12x = 9 + 6x \\ \\ =>6x = 45 \\ \\ => x = 9[/latex] Thus, the length of smaller peg is 9 cm.