# The perimeter of the rectangle is 56 and the diagonal is 27. Find the area of this rectangle.

June 26, 2021 | education

| Let’s denote the length of this rectangle through x, and the width of this rectangle through y.

Since the perimeter of this rectangle is 56, and its diagonal is 27, therefore, the following relations hold:

2x + 2y = 56;

x ^ 2 + y ^ 2 = 27 ^ 2.

From the first equation we get:

x + y = 56/2;

x + y = 28;

Let’s square both sides of the resulting ratio:

(x + y) ^ 2 = 28 ^ 2;

x ^ 2 + 2xy + y ^ 2 = 784;

2xy + x ^ 2 + y ^ 2 = 784.

Substituting the value x ^ 2 + y ^ 2 = 27 ^ 2 into the resulting ratio, we get:

2xy + 27 ^ 2 = 784;

2xy + 729 = 784;

2xy = 784 – 729;

2xy = 55;

xy = 55/2 = 27.5.

Therefore, the area of this rectangle is 27.5.

Answer: the area of this rectangle is 27.5.

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