Given a right-angled triangle ABC with a hypotenuse AB = 3√2. Find the length CA if sin A = 1/3

The sines of acute angles in a right-angled triangle are equal to the ratio of the opposite leg to the hypotenuse of the triangle. Hence,

sin A = BC / AB.

From this formula we express the leg BC:

BC = sin A * AB = 1/3 * 3√2 = √2.

According to the Pythagorean theorem in a right-angled triangle, the square of the hypotenuse is equal to the sum of the squares of its legs, that is:

AB² = BC² + CA².

From here

CA = √ (AB² – BC²) = √ ((3√2) ² – (√2) ²) = √ (9 * 2 – 2) = √ (18 – 2) = √16 = 4.

Answer: the length of the leg CA of the right triangle ABC is 4.