In a right-angled triangle ABC, the leg is AC = 65, and the height CH dropped to the hypotenuse is 13√21. 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:
AC2 = CH2 + AH2
65 ^ 2 = (13√21) 2 + AH ^ 2
4225 = 169 * 21 + AH ^ 2
AH ^ 2 = 4225-3549
AH ^ 2 = 676
AH = 26
sin∠ACH = AH / AC (by definition)
sin∠ACH = 26/65 = 0.4
As deduced above:
sin∠ABC = sin∠ACH = 0.4
Answer: sin∠ABC = 0.4