The area of a rectangle of a triangle is 180 cm2. Find the smaller leg of the triangle if it is 31 cm smaller than the other.

The area of ​​a right-angled triangle is half the product of its legs. S = 1/2 * ab, where a and b are legs of a triangle.

Let the larger leg a be equal to x cm, then the smaller leg b is equal to (x – 31) cm. By the condition of the problem, it is known that the area of ​​the triangle is 1/2 * x * (x – 31) cm² or 180 cm². Let’s make an equation and solve it.

1/2 * x * (x – 31) = 180;

x (x – 31) = 360;

x² – 31x – 360 = 0;

D = b² – 4ac;

D = (-31) ² – 4 * 1 * (-360) = 961 + 1440 = 2401; √D = 49;

x = (-b ± √D) / (2a);

x1 = (31 + 49) / 2 = 80/2 = 40 (cm) – larger leg a;

x2 = (31 – 49) / 2 = -18/2 = -9 – the length of the side of the triangle cannot be negative.

x – 31 = 40 – 31 = 9 (cm) – the length of the smaller side b.

Answer. 9 cm.