# A rectangle, one side of which is 2 cm larger than the other, has an area equal to the area of a square

A rectangle, one side of which is 2 cm larger than the other, has an area equal to the area of a square with a side 4 cm less than the perimeter of the rectangle. Find the sides of the rectangle.

1. Sides of rectangle a and b:

b = a + 2.

2. Perimeter of the rectangle:

P = 2 (a + b) = 2 (a + a + 2) = 2 (2a + 2) = 4 (a + 1).

3. Area of the rectangle:

S1 = ab = a (a + 2).

4. Side of the square:

x = P – 4 = 4 (a + 1) – 4 = 4a + 4 – 4 = 4a.

5. Square area:

S2 = x ^ 2 = (4a) ^ 2 = 16a ^ 2.

6. The areas of a rectangle and a square are equal:

S1 = S2;
a (a + 2) = 16a ^ 2;
a ^ 2 + 2a = 16a ^ 2;
15a ^ 2 – 2a = 0;
a (15a – 2) = 0;
15a – 2 = 0;
15a = 2;
a = 2/15;
b = a + 2 = 2/15 + 2 = 32/15.
Answer: 2/15 cm and 32/15 cm.

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