tan ACH = opp/adj = AH / CH
We know CH we just need AH.
We have three similar triangles. Each has a right angle and shares an acute angle with ABC, so similar by AA.
Let's get the order right,
ABC ~ HBA ~ HAC
It looks like from the last two we get an equal ratio equation involving AH, BH, and CH.
HBA ~ HAC
HB/HA = HA/HC
HB×HC = HA²
HA² = 4×8 = 32
HA = √32 = 4√2
tan ACH= AH / CH = 4√2 / 8 = √2/2
Answer: D) √2/2