Drawing a diagram of the triangle we have
Let a be the length of the perpendicular height of the triangle, then we can solve it using the Pythagorean Theorem
[tex]\begin{gathered} a^2+b^2=c^2 \\ a^2+(\frac{2x+4}{2})^2=(2x+1)^2 \\ a^2+(x+2)^2=(2x+1)^2 \\ a^2+x^2+4x+4=4x^2+4x+1 \\ a^2=4x^2+4x+1-x^2-4x-4 \\ a^2=3x^2-3 \\ \sqrt{a^2}=\sqrt{3x^2-3} \\ \end{gathered}[/tex]Therefore, the length of the perpendicular height of the triangle can be given in the expression
[tex]\sqrt{3x^2-3}[/tex]
