The length of a side of an equilateral triangle is the same as the length of a rectangle and the width of the rectangle is 2 inches less than its length, if the perimeter of the triangle s 4 inches less than the perimeter of the rectangle, what are the dimensions of the rectangle?

Question

anonymous · Answer

Help please!?

parthkohli · Answer

Let one side in the equilateral triangle be $x$.
As we know from the definition of perimeter, it's the sum of all sides. We do know that all sides are $x$ because it's an equilateral triangle.

parthkohli · Answer

The perimeter of that equilateral triangle is $x + x + x$ or $3x$, right?

anonymous · Answer

yes

parthkohli · Answer

Okay, well, now let's go to the rectangle.
We know that the length of the rectangle is the same as the side in our equilateral triangle.
The length becomes $x$ then, right?

anonymous · Answer

yes

parthkohli · Answer

OK, now we know that the width is 2 inches less than the length, so width becomes $x - 2$. Okay?

anonymous · Answer

yes

parthkohli · Answer

Do you know what the formula is for finding out the perimeter of a rectangle?

anonymous · Answer

2(l+w)

parthkohli · Answer

That's right! Now, we have to put in $x$ for length and $x - 2$ for width.$$\implies 2(x + x - 2) = Perimeter $$

anonymous · Answer

So, 2x+2x+4?

parthkohli · Answer

Wait! Let's just keep that like it is.$$2(x + x  - 2)  = 3x - 2 $$Agree with this?

anonymous · Answer

oh ok. yes

parthkohli · Answer

Actually, it'd be$$ 2(x + x - 2)  - 4 = 3x$$

parthkohli · Answer

Now solve the equation.

anonymous · Answer

2x+2x-4-4=3x
4x+8=3x
is that right so far?

parthkohli · Answer

No... :I

anonymous · Answer

No? how?

parthkohli · Answer

$$2(x + x - 2) - 4 = 3x$$Maybe simplify the parentheses.$$2(2x - 2) - 4 = 3x$$$$ 4x - 4 - 4 = 3x$$$$ 4x - 8 = 3x$$

parthkohli · Answer

As to your question, no, that's not the answer because when you complete solving that equation, you get $x = -8$ which is not possible. 

Why? There cannot be a negative side! Remember how $x = $ side of the triangle?

anonymous · Answer

oh ok.  whats next?

parthkohli · Answer

You get $x = 8$.

parthkohli · Answer

$x = \text{length of the rectangle }=8\ x -2 = \text{width of the rectangle} = 8 - 2 = 6$

anonymous · Answer

wait can we go back to 4x-8=3x. how did you get from there to x=8?

parthkohli · Answer

$$\begin{array}{lr} 4x - 8 &=&3x\ 4x - 8 + 8&=&3x + 8\ 4x &=&3x + 8\4x - 3x&=&3x + 8 - 3x\ x &=&8 \end{array}$$

anonymous · Answer

oh ok. thats what i thought just checking.

anonymous · Answer

whats next?

parthkohli · Answer

I wrote it above.

anonymous · Answer

so the width of the rectangle is 6 and the length is 8?