SOLUTION: Geometry. The length of a rectangle is 1 cm longer than its width. If the diagonal of the rectangle is 4 cm, what are the dimensions (the length and the width) of the rectangle?

Question 117871: Geometry. The length of a rectangle is 1 cm longer than its width. If the diagonal
of the rectangle is 4 cm, what are the dimensions (the length and the width) of the rectangle?

Found 2 solutions by ankor@dixie-net.com, solver91311:
Answer by ankor@dixie-net.com(22740) (Show Source): You can put this solution on YOUR website!
The length of a rectangle is 1 cm longer than its width. If the diagonal
of the rectangle is 4 cm, what are the dimensions (the length and the width) of the rectangle?
:
Let x = the width
then
(x+1) = the length
:
The sides and the diagonal from a right triangle so we can use pythagorus here
a^2 + b^2 = c^2
:
In our problem a = x; b = (x+1); c = 4
:
x^2 + (x+1)^2 = 4^2
:
FOIL (x+1)(x+1)
x^2 + x^2 + 2x + 1 = 16
:
2x^2 + 2x + 1 - 16 = 0
:
2x^2 + 2x - 15 = 0; a quadratic equation
:
Solve for x using the quadratic formula; a=2; b=2; c=-15

:

:

:
; minus a minus is a plus
:
; we only want the positive solution here
:

:
x = 2.284 cm is the width
then
2.284 + 1 = 3.284 cm is the length
:
:
Check our solution:
2.284^2 + 3.284^2 =
5.2 + 10. 8 = 16 which is 4^2
Answer by solver91311(24713) (Show Source): You can put this solution on YOUR website!
If you create a diagonal in a rectangle, then you have created a right triangle with the diagonal as a hypotenuse. Let's say the short leg of the triangle is x, and then the long leg (the long side of the rectangle) has to be x + 1. And we are given that the hypotenuse is 4 cm.
Pythagoras tells us that the sides and hypotenuse of a right triangle are related thus:

, or .

Replace a, b, and c with the values we are given:

Expand the binomial and combine terms under the radical

Square both sides:

Add -16 to both sides:

Use the quadratic formula

, so

, or
.

so we can exclude this answer because we are looking for a positive number length.

Since , , so is the answer we are seeking and is the width of the rectangle. The length of the rectangle is , so the length must be

Hope this helps,
Happy Holidays,
John

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