The length of AC will be 2.5495.... cm.
Explanation
In the diagram below, ABC is a right angle triangle with altitude as CD.
So, triangle ADC will be also a right angle triangle, in which CD = 2 cm.
In triangle ADC, using Pythagorean theorem we will get.....
[tex]AD^2 +CD^2= AC^2 \\ \\ AD^2+(2)^2= AC^2\\ \\ AD^2= AC^2 -4 \\ \\ AD= \sqrt{AC^2 -4} ..............................(1)[/tex]
Now in triangle ABC, using Pythagorean theorem.....
[tex]AC^2 + BC^2= AB^2\\ \\ AC^2+ BC^2= (3)^2\\ \\ BC^2= 9-AC^2\\ \\ BC= \sqrt{9-AC^2}......................................(2)[/tex]
As it is given that AD = BC , so from equation (1) and (2) we will get.....
[tex]\sqrt{AC^2 -4}=\sqrt{9-AC^2}\\ \\ AC^2-4= 9-AC^2\\ \\ 2AC^2= 9+4=13\\ \\ AC^2= \frac{13}{2}=6.5\\ \\ AC= \sqrt{6.5}=2.5495....[/tex]
So, the length of AC will be 2.5495.... cm.
