The area of a right-angled triangle is (50√3) / 3. One of the sharp corners is 60. Find the length of the leg opposite this corner.

Triangle ABC – rectangular, angle B = 90 degrees, angle C = 60 degrees.
The area of ​​a right-angled triangle can be determined through the hypotenuse and acute angle:
S = (c ^ 2 * sin2α) / 4.
1. Find the length of the hypotenuse:
(50√3) / 3 = (c ^ 2 * sin (2 * 60)) / 4;
(50√3) / 3 = (c ^ 2 * sin120) / 4;
(50√3) / 3 = (c ^ 2 * (√3 / 2)) / 4;
(50√3) / 3 = √3c ^ 2/8;
3√3c ^ 2 = 400√3 (proportional);
c ^ 2 = 400√3 / 3√3;
c ^ 2 = 400/3;
c = √ (400/3);
c = 20 / √3;
c = 20√3 / 3 conventional units.
Hypotenuse AC = 20√3 / 3 conventional units.
2. Angle A + angle B + angle C = 180 (according to the theorem on the sum of the angles of a triangle);
angle A + 90 + 60 = 180;
angle A = 180 – 150;
angle A = 30 degrees.
The leg BC lies against the angle A, which is 30 degrees, therefore:
BC = AC / 2;
BC = (20√3 / 3) / 2 = 20√3 / 6 = 10√3 / 3 (conventional units).
3. By the Pythagorean theorem:
AB = √ (AC ^ 2 – BC ^ 2);
AB = √ (20√3 / 3) ^ 2 – (10√3 / 3) ^ 2) = √ (1200/9 – 300/9) = √ ((1200 – 300) / 9) = √ (900 / 9) = √100 = 10 (conventional units).
Answer: AB = 10 conventional units.