The perimeter of the rectangle is 18 and the diagonal is √41. Find the area of this rectangle.
May 30, 2021 | education
| 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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