In a right-angled triangle ABC, angle C is straight. Find BC if sinA = 3/4, AC = 8√7.

Let us denote the hypotenuse AB by y, and the unknown leg BC by x.

Then, by the Pythagorean theorem for a right-angled triangle, the following equality will be true:

y ^ 2 = x ^ 2 + (8 * √7) ^ 2;

y ^ 2 = x ^ 2 + 64 * 7;

y ^ 2 = x ^ 2 + 448;

The sine in a right-angled triangle is the ratio of the opposite leg to its hypotenuse, i.e. sin A = BC / AB;

Thus, we get:

3/5 = x / y;

We express x and substitute in the first equality:

x = 3/5 * y;

y ^ 2 = (3/5 * y) ^ 2 + 448;

y ^ 2 = 9/25 * y ^ 2 + 448;

y ^ 2 – 9/25 * y ^ 2 = 448;

16/25 * y ^ 2 = 448;

y ^ 2 = 448 * 25/16;

y = √448 * √25 / 16 = √ (4 * 112) * 5/4 = 2 * √ (4 * 28) * 5/4 = 2 * 5/4 * 2 * √28 = 5 * √ (4 * 7) = 5 * 2 * √7 = 10 * √7.

x = 3/5 * 10 * √7 = 6 * √7.

Answer: 6 * √7.