The rectangle has a perimeter of 28 cm and its diagonal is 10 cm. Find the area of the rectangle.

1. Vertices of the rectangle A, B, C, D. BD is the diagonal.

2. Let us denote the length of the side AD by the index “a”, the length of the side AB by the index “b”.

3. Let’s compose two equations:

(1) 2a + 2b = 28 (formula for calculating the perimeter of a rectangle); a + b = 14; a = (14 – c);

(2) a² + b² = 10² (according to the Pythagorean theorem from the right-angled triangle AED); a² + b² = 100;

4. Substitute the value a = (14 – b) into the second equation:

(14 – c) ² + b² = 100;

196 – 28v + b² + b² = 100;

2v² – 28v + 96 = 0;

в² – 14в + 48 = 0;

The first value in = (14 + √196 – 4 x 48) / 2 = (14 + 2) / 2 = 8 cm.

The second value is in = (14 – 2) / 2 = 6 cm.

The first value is a = 14 – 8 = 6 cm.

The second value is a = 14 – 6 = 8 cm.

5. The length of the AD side can be 8 cm or 6 cm.

The length of the AB side can be 6 cm or 8 cm.

4. Calculate the area (S) of a given rectangle:

S = ab = 6 x 8 = 48 cm².

Answer: the area of ​​a given rectangle is 48 cm².