A rectangle has area 6 cm^2 and diagonal length 2 square root 5cm. What is its perimeter

Answer 1

Let the length and width be x and y, both x and y positive

xy = 6 and x^2 + y^2 = (2√5)^2

from xy=6, y = 6/x

x^2 + (6/x)^2 = 20
x^2 + 36/x^2 = 20
x^4 + 36 = 20x^2
x^4 - 20x^2 + 36 = 0
(x^2 - 18)(x^2 - 2) = 0

x^2 = 18
x = √18 = 3√2, and y = 6/√18 = √2
similarly if x^2 = 2 , then x = √2 and y = 3√2

the perimeter = 2x + 2y
= 6√2 + 2√2 = 8√2

check:
area = (3√2)(√2) = 3(2) = 6 , as given
diagonal:
d^2 = x^2 + y^2
= 18 + 2
= 20
= 2√5 , as given

Answer 2

Find x if the area of the figure shown is A=2 square root of 6cm square