In a right-angled triangle, one of its angles is 60 degrees. The sum of the hypotenuse and the smaller

In a right-angled triangle, one of its angles is 60 degrees. The sum of the hypotenuse and the smaller leg is 45 √3. Find a larger leg.

If one angle is 90 °, the other is 60 °, then the third angle is 30 °. It is the smallest, and opposite it will be the smallest leg. The leg opposite an angle of 30 ° is equal to half of the hypotenuse, that is, (45√3) / 2.

Let the smaller leg be x, then the hypotenuse is equal to 2x.

The sum of the lesser leg and the hypotenuse:

x + 2x = 45√3.

3x = 45√3.

Smaller leg: x = 15√3.

Hypotenuse: 2x = 30√3.

Let the large leg be equal to y. The sum of the squares of the legs is equal to the square of the hypotenuse:

y2 + (15√3) 2 = (30√3) 2;

y2 = 900 * 3 – 225 * 3 = 3 * 675 = 2025;

y = √2025 = 45.

Answer: The larger leg is 45.