Answer to Question #264696 in Geometry for Ali

Question

The area of a rectangle is: x²-1/2

a)What must be the length of this rectangle if its width is x-1

b)For what value of x is this rectangle a square?

Answer 1

We know that Area of a rectangle is

$Area=(length*breadth)unit²$

$A=l*b$

Where $A=x²-½$

(a) we have width, $b$ to be:

$b=x-1$

Let the length be $y$

$A=l*b$

$y(x-1)=x²-½$

Dividing both sides by $x-1$

$y=\frac{x²-½}{x-1}$

$y=\frac{2x²-1}{2x-1}$ unit

Which gives the expression for the length of the rectangle as $y$ inform of $x$.

(b) We have the formula for Area of a square to be:

$A=l²$ (in unit squared)

Where $l=x-1$

$l²=A$

$(x-1)²=x²-½$

$x²-2x+1=x²-½$

By collecting the like terms

$x²-x²-2x=-½-1$

$-2x=-\frac{3}{2}$

Multiply through by -1

$2x=\frac{3}{2}$

Cross multiply

$4x=3$

$x=\frac{3}{4}$

Therefore, the value of $x$ for which the rectangle is a square is $\frac{3}{4}$