If the width of a rectangle is 1 inch less than its length?

If Length is L(say x inches ) and Breadth is B( say x-1) and Diagonal is D( say x+1)
then L^2+B^2=D^2;
x^2+(x-1)^2 =(x+1)^2
x^2-4x=0

=> x=4 inches

The theorum of pythagorus states that the square root of the sum of the square of the two sides of a right angled triangle is equal to the diagonal line.
Thus, assuming that A is the length, B is the width and C is the diagonal of such a triangle:
C = sqrt(A^2 + B^2)
So, assuming the length of the triangle is x:
x +1 = sqrt((x-1)^2 + x^2)
If you solve for x, you'll find that:
x = 4 inches (length) and the width = 3 inches

x = length

x^2 + (x - 1)^2 = (x + 1)^2
x^2 + (x - 1)(x - 1) = (x + 1)(x + 1)
x^2 + x^2 - x - x + 1 = x^2 + x + x + 1
2x^2 - x^2 - 2x - 2x + 1 - 1 = 0
x^2 - 4x = 0
x(x - 4) = 0

x = 0

x - 4 = 0
x = 4
(the length should be 4 inches)

x - 1
= 4 - 1
= 3

∴ the length should be 4 inches and the width should be 3 inches.

l is width, the length l+1, the diagonal l+2
using pytagoras
(l+2)^2=(l+1)^2+l^2
l^2+4l+4=l^2+2l+1+l^2=2l^2+2l+1
0=l^2-2l-3
2 roots l=3 and l=-1imposible
so Answer width 3 , length 4, diagonal 5

Let the length be x.

Length = x inches
Width = (x - 1) inches
Diagonal = (x + 1) inches

The three lines form a right-angled triangle.

Using Pythagoras Theorem:

(x + 1)^2 = x^2 + (x - 1)^2
x^2 + 2x + 1 = 2x^2 - 2x + 1
x^2 - 4x = 0
x(x - 4) = 0
Hence x = 0 or x = 4

Therefore length = x = 0 or 4, width = x -1 = -1 or 3

Since the length cannot be 0 and the width cannot be -1 inch, we can only conclude that:

length of rectangle = 4 inches & width of rectangle = 3 inches.

width = length - 1
diagonal = length + 1

diagonal^2 = length^2 + width^2 (pythagorean theorem)

By substituting the first 2 equations into the third one you get:
(length+1)^2 = length^2 + (length-1)^2

Expand it:
length^2 + 2*length + 1 = length^2 + length^2 - 2*length + 1

Simplify:
length^2 + 2*length + 1 = 2*length^2 - 2*length + 1
4*length = length^2
4 = length
width = length - 1 = 4 - 1 = 3

So:
length = 4 inches
width = 3 inches

(And though you didn't ask for it, the diagonal is 5 inches, which makes a 3-4-5 right triangle. This makes sense, as there has to be a right angle there for it to be a rectangle.)