The hypotenuse of a right-angled triangle is 10 more than one leg and 20 more than the other. Find the area of this triangle.

We take the hypotenuse of the triangle for x, then:
x-10 – length of one leg,
x-20 is the length of the other.
By the Pythagorean theorem, we obtain the equation:
(x-10) ^ 2 + (x-20) ^ 2 = x ^ 2
x ^ 2-20x + 100 + x ^ 2-40x + 400 = x ^ 2
x ^ 2-60x + 500 = 0
x12 = (60 + -√3600-4 * 500) / 2 = (60 + 40) / 2 = 50
The legs are equal:
50-20 = 30
50-10 = 40
Then the area is equal to:
30 * 40 * 1/2 = 1200/2 = 600.