The square and the rectangle have the same area. what is the numercal value of the rectangle's perimeter?

Question

value of one side of the square: x

value of length of rectangle: 2x-3

value of width of rectangle: x-2

Answer 1

You have your sides in algebraic expression.  All you need to do is apply the area of a square and rectangle to them to establish an equation.

 

Area of square = area of rectangle

 

x² = (2x - 3)(x - 2)

 

 

Expand the right side of the equation using FOIL.

 

x² = 2x² - 7x + 6

 

 

Subtract x² on both sides of the equation to make the left side equal to zero.  Doing this will allow us to easily use FOIL.

 

0 = x² - 7x + 6

 

 

Factor the right side.

 

0 = (x + 6)(x + 1)

 

 

Set the factors equal to zero and solve for x from each.  Then chose the most reasonable value of x.

Answer 2

Since one side of the square is x, the area of the square will be:

 

area of square = x²

 

The area of the rectangle is l * w, so:

 

area of rectangle = (2x - 3)(x - 2) = 2x² - 7x + 6

 

Since the two areas are equal to each other:

 

x² = 2x² - 7x + 6

 

0 = x² - 7x + 6

 

Factor the quadratic:

 

0 = (x - 6)(x - 1)

 

So, either:

 

x - 6 = 0 or

 

x - 1 = 0 or

 

both are equal to 0.

 

x - 6 = 0 => x = 6

 

x - 1 = 0 => x = 1

 

x can't be equal to 1 since that would make the rectangle's length and width negative.

 

So, x must be 6.

 

To find the perimeter of the rectangle, substitute 6 for x in the length and width expressions:

 

length = 2(6) - 3 = 9

 

width = 6 - 2 = 4

 

The perimeter then is:

 

2(9) + 2(4) = 18 + 8 = 26