Answer:
diagonal ≈ 18.43 cm
Step-by-step explanation:
Let L represent the length of the rectangle. Then the width is ...
w = 2L -4 . . . . . . 4 less than twice the length
The area is ...
A = wL = (2L -4)L = 2L² -4L
The area is said to be 153 cm², so we have ...
2L² -4L = 153
2L² -4L -153 = 0 . . . . . . subtract 153 to put into standard form
We can find the solution to this using the quadratic formula. It tells us the solution to ax²+bx+c=0 is given by ...
x = (-b±√(b²-4ac))/(2a)
We have a=2, b=-4, c=-153, so our solution for L is ...
L = (-(-4) ±√((-4)²-4(2)(-153)))/(2(2)) = (4±√1240)/4
Only the positive solution is of interest, so L = 1+√77.5.
__
Now we know the rectangle is 1+√77.5 long and -2+2√77.5 wide. The diagonal (d) is the hypotenuse of a right triangle with these leg lengths. Its measure can be found from ...
d² = w² +L² = (-2+2√77.5)² +(1+√77.5)²
It can work well to simply evaluate this using a calculator, or it can be simplified first.
d² = 4 -8√77.5 +4·77.5 + 1 +2√77.5 +77.5 = 392.5 -6√77.5
Taking the square root gives the diagonal length:
d = √(392.5 -6√77.5) ≈ 18.43 . . . . cm
