# 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

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:

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

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