In triangle ABC, angle C is 90 degrees, angle A is 30 degrees, AC = 19√3. Find BC.

We find the leg AC using the Pythagorean theorem:

AB ^ 2 = AC ^ 2 + CB ^ 2;

AC = √ (AB ^ 2 – CB ^ 2);

CB = 1/2 AB (CB – the leg, lying opposite an angle of 30 ° is equal to half of the hypotenuse)

AB = 2 * CB;

AC = √ (4 * CB ^ 2 – CB ^ 2) = CB * √3;

19√3 = CB * √3;

Answer: Leg CB = 19.

Note. You can solve the problem using trigonometric functions:

CB / AC = tg 30 °;

CB = AC * tan 30 ° = 19√3 * (√3 / 3) = 19.