In a right-angled triangle, the projection of the smaller leg onto the hypotenuse is 2 cm, the hypotenuse is 18 cm. Find the smaller leg.

Let us prove that the triangles ABН and BCH are similar.
Let the angle BCH = X0, then the angle BCH = (90 – X) 0.
In triangle ABН, angle ABН = (90 – СВН) = (90 – (90 – X) = X0.
Then the triangles ABН and СВН are similar in acute angle.
Then, in similar triangles:
ВН / AН = CH / ВН.
BH ^ 2 = AH * CH = 2 * 8 = 16 cm.
BH = 4 cm.
Since the projection of the BC leg is less than the projection of the AB leg, the BC is the smaller leg of the triangle.
In a right-angled triangle ABC, according to the Pythagorean theorem, BC2 = BH2 + CH2 = 16 + 4 = 20.
BC = √20 = 2 * √5 cm.
Answer: The length of the smaller leg is 2 * √5 cm.