How do you find the width and length of a rectangle knowing ONLY the diagonal and the area?

Answer 1

For e.g we draw a rectangle with length and width of 4cm and 3cm respectively.

So the diagonal would be of 5cm(using Pythagoras theorem) and area 12cm^2.

Now using algebra we can find the dimensions of the rectangle,Look below:

let length=x

let width=y

so: x*y=12----(1)

x^2+y^2=5^2----(2)

x^2+y^2=25

Now simplifying the above equations:

x^2=25-y^2

---->x^2*y^2=144

---->(25-y^2)*y^2=144

25y^2-y^4=144

y^4-25y^2+144=0

let y^4=z^2

so:

z^2-25z+144=0

by simplifying the above quadratic equation:

z=(25+(25^2-4(1)(144))^(1/2))/2(1) , z= (25-(25^2-4(144))^(1/2))/2

z=16, z=9

when z=16, when z=9

so y=4 ,so y=3

Now verifying which magnitude is correct for y:

12/4=x ,12/3=x

x=3 not correct, because y>x x=4 is correct, because x>y