ΔAED is the equilateral triangle. Therefore AH is a height of this triangle.
The formula of a height of an equilateral triangle:
[tex]h=\dfrac{a\sqrt3}{2}[/tex]
We have a = 5. Substitute:
[tex]\boxed{AH=\dfrac{5\sqrt3}{2}}[/tex]
You can use the Pythagorean theorem.
[tex]EH^2+AH^2=AE^2\\\\\left(\dfrac{5}{2}\right)^2+AH^2=5^2\\\\\dfrac{25}{4}+AH^2=25\\\\AH^2=25-\dfrac{25}{4}\\\\AH^2=\dfrac{100}{4}-\dfrac{25}{4}\\\\AH^2=\dfrac{75}{4}\to AH=\sqrt{\dfrac{75}{4}}\\\\AH=\dfrac{\sqrt{75}}{\sqrt4}\\\\AH=\dfrac{\sqrt{25\cdot3}}{2}\\\\\boxed{AH=\dfrac{5\sqrt3}{2}}[/tex]
