The Solution:
Given:
We are required to find x.
By the Pythagorean Theorem:
[tex]\begin{gathered} (x+6)^2=x^2+(x+3)^2 \\ x(x+6)+6(x+6)=x^2+x(x+3)+3(x+3) \end{gathered}[/tex][tex]\begin{gathered} x^2+6x+6x+36=x^2+x^2+3x+3x+9 \\ \text{ Collecting the like terms, we get:} \\ x^2-2x^2+12x-6x+36-9=0 \\ -x^2+6x+27=0 \\ Multiplying\text{ through with -1, we get} \\ x^2-6x-27=0 \end{gathered}[/tex]
Solving as a quadratic equation, we get:
[tex]\begin{gathered} x^2-9x+3x-27=0 \\ x(x-9)+3(x-9)=0 \\ (x+3)(x-9)=0 \\ x+3=0 \\ x=-3\text{ \lparen discard because it is negative.\rparen} \\ x-9=0 \\ x=9cm \end{gathered}[/tex]
Therefore, the correct answer is 9cm
