In a right-angled triangle ABC, the leg is AC = 35, and the height CH dropped to the hypotenuse is 14√6. Find sin∠ABC

Consider the triangles ABC and ACH.
∠AHC = ∠ACB (as these are right angles).
∠A is common.
Therefore, by the theorem on the sum of the angles of a triangle, ∠ACH = ∠ABC
Then sin∠ACH = sin∠ABC.
Now consider the triangle ACH.
By the Pythagorean theorem:
AC ^ 2 = CH6 ^ + AH ^ 2
35 ^ 2 = (14√6) ^ 2 + AH ^ 2
1225 = 196 * 6 + AH ^ 2
AH ^ 2 = 1225-1176
AH ^ 2 = 49
AH = 7
sin∠ACH = AH / AC (by definition)
sin∠ACH = 7/35 = 1/5 = 0.2
As deduced above:
sin∠ABC = sin∠ACH = 0.2
Answer: sin∠ABC = 0.2