In ΔABC, AC = BC, CD ⊥ AB with D ∈ AB , AB = 4 in, and CD = sqrt3 in. Find AC.

check the picture below.

segments AC and BC are equal, that means that ABC is an isosceles triangle, with twin sides.  Now the twin sides make twin angles on the opposite side, namely the angles at A and B are twins.

if the angles at A and B are twins, and CD is ⟂ to AB, there's only one possibility that can happen, and is if CD is an angle bisector at C.

an angle bisector like CD with twin angles on each side, will cut AB in two equal halves, therefore, if AB = 4, then AD = 2 and DB = 2.