Calculate the height AH in the right triangle ABC

5. ABC dik üçgen, $[AH] \perp [BC]$, $[AB] \perp [AC]$, $|AD| = |AC|$, $|CD| = 4|BD|$ ve $|BD| = 2$ cm'dir.

[Görselde bir ABC üçgeni var. A noktasında dik açı, AH kenarı BC'ye dik. AD ve AC kenarları üzerinde çift çizgi ile eşitlik işareti var.]

Buna göre $|AH|$ kaç cm'dir?

A) $2\sqrt{6}$

B) $3\sqrt{5}$

C) $4\sqrt{3}$

D) $5\sqrt{2}$

E) $6$

Soruda görsel içerik var: The image shows a triangle ABC with a right angle at A (labeled with a square symbol). A line segment AH is drawn from vertex A perpendicular to base BC (labeled with a square symbol at H). D is a point on the line segment BH. There is a line AD. Equality markings are placed on segments AD and AC, indicating that they are equal in length. The vertices B, D, H, C lie on a straight line in that order.