The perimeter of the rectangle is 196 and the diagonal is 82. Find the length of the smaller side of this rectangle.

The square of the diagonal of the rectangle is equal to the sum of the squares of the adjacent sides:

a ^ 2 + b ^ 2 = 82 ^ 2 = 6724.

The sum of the lengths of two adjacent sides of a rectangle is half of its perimeter:

a + b = 196/2 = 98.

Let’s square both sides of this equality:

(a + b) ^ 2 = 98^2;

a ^ 2 + b ^ 2 + 2ab = 9604;

6724 + 2ab = 9604;

2ab = 9604 – 6724 = 2880;

ab = 2880/2 = 1440.

We get the system of equations:

1) a + b = 98,

2) ab = 1440.

From the first equation: a = 98 – b, substitute into the second equation:

(98 – b) * b = 1440;

98b – b ^ 2 = 1440;

b ^ 2 – 98b + 1440 = 0.

D = 98 ^ 2 – 4 * 1440 = 9604 – 5760 = 3844 = 622;

b1 = (98 – 62) / 2 = 36/2 = 18,

b2 = (98 + 62) / 2 = 160/2 = 80.

The smaller side of this rectangle is 18.