# 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

May 31, 2020 | Education

| 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

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