Explanation:
geometric mean theorem for right-angled triangles :
height = sqrt(pq)
where p and q are the parts of side as split by the height of that side.
19.
x = sqrt(4×16) = sqrt(64) = 8
20.
y = sqrt(5×8) = sqrt(40) = 6.32455532...
21.
18 = sqrt(12y)
18² = 12y
324 = 12y
y = 27
22.
10 = sqrt(25x)
10² = 25x
100 = 25x
x = 4
23.
Pythagoras
x² = 5² + height² = 25 + 4×5 = 25 + 20 = 45
x = sqrt(45) = 6.708203932...
24.
Pythagoras
b² = 16² + height² = 256 + 6×16 = 256 + 96 = 352
b = sqrt(352) = 18.76166304...
25.
Pythagoras
27² = 16² + height² = 256 + 16(z - 16) =
= 256 + 16z - 256 = 16z
729 = 16z
z = 729/16 = 45.5625
26.
Pythagoras
x² = (8 - 2)² + height² = 6² + (8 - 2)×2 =
= 36 + 12 = 48
x = sqrt(48) = 6.92820323...