One leg of a right-angled triangle is 34 larger than the other, and the hypotenuse is 106. Find the area of the triangle.

One leg of a right-angled triangle is 34 larger than the other, and the hypotenuse is 106. Find the area of the triangle.

Let x be the length of one leg, then x + 34 is the length of the other leg.

By the Pythagorean theorem, we compose the equation:

X² + (x + 34) ² = 106²;

x² + x² +68 * x + 34² = 106²;

2 * x² + 68 * x + 1156 = 11236;

2 * x² + 68 * x + 1156 – 11236 = 0;

2 * x² + 68 * x – 10800 = 0; Divide both sides of the equation by 2;

x² + 34 * x – 5400 = 0;

We solve this equation:

x1 = – 90, which does not fit according to the problem statement;

x2 = 56;

56 – the first leg;

56 + 34 = 90 – the second leg.

Answer: 56 and 90.