Explanation:
first of all we need to understand the given information.
2 shapes are similar, when they have the same angles, AND when every pair of corresponding sides (or side lengths) or any other lines like heights or diagonals has the same scaling factor.
A B/X Y = 4 is the scaling factor.
it means that the large side A B is 4 times a long as the corresponding short side X Y in the pair A B, X Y.
and the same goes for e.g. B C/Y Z and all the other side or length pairs - every ratio is 4.
so, in fact, to get the measurements of the small shape we need to multiply the measurements of the large shape by 1/4.
it is not given information, but I assume it is a symmetric hexagon with A B = E D, A F = F E = B C = C D.
and as both hexagons are similar, the same applies to the corresponding sides in the small hexagon.
for the area of ABCDEF we can split the hexagon into 2 equal trapezoids ABCF and EDCF.
the area of such a symmetrical trapezoid is
(baseline + topline)/2 × height
baseline = C F = 10 cm
topline = A B = 5 cm
height = A E/2 = 12/2 = 6 cm
so, the area of the whole hexagon is
ABCDEF = 2×(10+5)/2 × 6 = 15 × 6 = 90 cm²
W Z = C F × 1/4 = 10/4 = 2.5 cm
U V = X Y = A B × 1/4 = 5/4 = 1.25 cm
U Y = A E × 1/4 = 12/4 = 3 cm
the height of a small trapezoid is then U Y/2 = 3/2 = 1.5 cm
to get the shaded area we need to calculate the area of the small hexagon and subtract it from the area of the large hexagon :
UVWXYZ = 2×(2.5 + 1.25)/2 × 1.5 = 3.75 × 1.5 = 5.625 cm²
so, the shaded area is
90 - 5.625 = 84.375 cm² ≈ 84.38 cm²