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

First, let’s remember the formula for the perimeter of a rectangle.
P = 2 * (a + b), where a – length, b – width.

So, P = 2 * (a + b) = 28;

Hence:

a + b = 28/2 = 14;

Next, recall that the diagonal (c) of a rectangle with sides a and b divides it into 2 equal right-angled triangles, where a and b are legs, c is the hypotenuse.
By the Pythagorean theorem, we find the legs:
a ^ 2 + b ^ 2 = c ^ 2;

a ^ 2 + b ^ 2 = 10 ^ 2;

So, we got a system of equations:

a + b = 14;

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

From the first equation, we express b through a:
b = 14 – a;

In the second equation, replace b with 14 – a:
a ^ 2 + (14 – a) ^ 2 = 10 ^ 2;

Let’s square 14 – a by the formula:
(a – b) ^ 2 = a ^ 2 – 2ab + b ^ 2;

(14 – a) ^ 2 = 196 – 28a + a ^ 2;

Hence:

a ^ 2 + 196 – 28a + a ^ 2 = 100;

Let’s simplify the expression:
2 * a ^ 2 – 28 * a + 196 = 100;

Move 100:
2 * a ^ 2 – 28 * a + 196 – 100 = 0;

2 * a ^ 2 – 28 * a + 96 = 0;

Let’s shorten the expression by dividing each term by 2:
a ^ 2 – 14 * a + 48 = 0;

So we got a quadratic equation.

Let’s define the discriminant:
D = (-14) ^ 2 – 4 * 48 = 196 – 192 = 4 = 2 ^ 2;

Let’s find a:
a1 = (14 – 2) / 2 = 12/2 = 6;

a2 = (14 + 2) / 2 = 16/2 = 8;

Find b:
b = 14 – a = 14 – 6 = 8.

So, knowing the length and width of the rectangle (8 and 6), we find its area:
S = a * b;

S = 8 * 6 = 48 cm ^ 2;

Answer: 48 cm ^ 2.