# Find the diagonal of a rectangle if its perimeter is 58 cm and its area is 120 cm2.

Let’s write a formula for calculating the perimeter of a rectangle in terms of its length and width:

Denoting the length – a, width – b.

P = 2 * (a + b).

Substitute in the known perimeter value and express the length of the rectangle:

58 = 2 * (a + b).

a + b = 58/2.

a + b = 29.

a = 29 – b.

Formula for the area of a rectangle in terms of its sides:

S = a * b.

Substitute in the known area and also express the length of the rectangle:

120 = a * b.

a = 120 / b.

Let’s equate the values of the lengths through the width:

29 – b = 120 / b.

(b ^ 2) – 29 * b + 120 = 0.

Let’s solve the resulting quadratic equation:

D = (- 29 ^ 2) – 4 * 1 * 120 = 841 – 480 = 361 = 19 ^ 2.

b1 = (29 – 19) / 2 = 5 cm.

b2 = (29 + 19) / 2 = 24 cm.

Let us now find the width of the rectangle using any of the formulas. Let’s use the following formula:

a = 29 – b.

Substitute the found length values:

a1 = 29 – b1.

a2 = 29 – b2.

a1 = 29 – 5 = 24 cm.

a2 = 29 – 24 = 5 cm.

As you can see, the obtained values of length and width are mutually inverse. Length 24 cm, width 5 cm.

To find the diagonal, use the formula (a ^ 2) + (b ^ 2) = c ^ 2.

Substitute the found values into the formula:

(24 ^ 2) + (5 ^ 2) = 576 + 25 = 601 = 24.5 cm.

The diagonal of the rectangle is 24.5 cm.

Answer: 24.5 cm.