Solution:
Given the triangle below where K is in the incenter:
Thus, we have
step 1: In the triangle DGK, find DK.
Thus, by Pythagoras theorem, we have
[tex]DK=\sqrt{18^2+(9x-16)^2}[/tex]
step 2: In the triangle KDI, find DK.
Similarly, we have
[tex]DK=\sqrt{18^2+(4x+9)^2}[/tex]
Step 3: Equate the equations in steps 1 and 2.
This gives
[tex]\begin{gathered} \sqrt{18^2+(9x-16)^2}=\sqrt{18^2+(4x+9)^2} \\ take\text{ the square of both sides,} \\ \Rightarrow18^2+(9x-16)^2=18^2+(4x+9)^2 \\ thus,\text{ we have} \\ 9x-16=4x+9 \\ add\text{ -4x to both sides,} \\ 9x-4x-16=-4x+4x+9 \\ 5x-16=9 \\ add\text{ 16 to both sides,} \\ 5x-16+16=9+16 \\ \Rightarrow5x=25 \\ divide\text{ both sides by the coefficient of x, which is 5} \\ \frac{5x}{5}=\frac{25}{5} \\ \Rightarrow x=5 \end{gathered}[/tex]
The value of x is
[tex]5[/tex]
