# The perimeter of the rectangle is 18 and the diagonal is √41. Find the area of this rectangle.

1. The perimeter of the rectangle is P = 2 * (a + b). Let us express “b” through “a”:

18 = 2 * (a + b);

a + b = 9;

b = 9 – a.

2. The diagonal of a rectangle divides it into two equal right-angled triangles and is a hypotenuse (c). The sides of the rectangle are legs (a, b). By the Pythagorean theorem, we compose and solve the equation:

a ^ 2 + b ^ 2 = c ^ 2;

a ^ 2 + (9 – a) ^ 2 = (√41) ^ 2;

a ^ 2 + 81 – 18a + a ^ 2 = 41;

2a ^ 2 – 18a + 40 = 0;

a ^ 2 – 9a + 20 = 0;

D = 81 – 80 = 1;

a1 = (9 + 1) / 2 = 5;

a2 = (9 – 1) / 2 = 4;

b1 = 9 – 5 = 4;

b2 = 9 – 4 = 5.

3. Find the area of the rectangle:

S = a * b = 5 * 4 = 20 (unit2).

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