# 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.