To find the perimeter of the rectangle, add all four sides together. Even though we are not given the length of all four sides, we know that opposite sides are always equal. 

\(\displaystyle 12 + 2 + 12 + 2 = 28\)

In order to figure out how many feet of fence you need, add up all of the segments that are going to be fenced.  Thus, \(\displaystyle 12+12+18+18+36= 92\).  Remember, you will not need to put fence where the house is.

Since a rectangle has two sets of equal sides, the perimeter is the sum of twice each of the two sides given. \(\displaystyle 2\cdot14=28\) and \(\displaystyle 2\cdot27=54\). \(\displaystyle 28+54=82\), which is the answer.

Since rectangles have two sets of equal sides, and one side is of length \(\displaystyle 12\), there must be another side of length \(\displaystyle 12\). Since we know the total perimeter of the rectangle, we can get the length of the two remaining sides by subtracting the two known side lengths from the total perimeter. \(\displaystyle 40-12-12=16\), and since the rectangle must have another set of equal sides, we must split \(\displaystyle 16\) into two equal parts. Therefore, the last two sides must be \(\displaystyle 8\) and \(\displaystyle 8\). This means the rectangle's other three sides must be \(\displaystyle 12\), \(\displaystyle 8\), and \(\displaystyle 8\).

The perimeter is the measurements of all four sides of a square or rectangle. We can use the following formula to solve the problem:

\(\displaystyle \textup{Perimeter} = 2 (\textup{length}) + 2 (\textup{width})\)

We already know that one side is 15 ft and that the total perimeter is 76 ft, so we can plug those numbers into the formula.

\(\displaystyle 76 = 2 (15) + 2 (\textup{w})\)

\(\displaystyle 76 = 30 + 2 (\textup{w})\)

We can subtract 30 from both sides of the equation.

\(\displaystyle 76-30 = 30-30 + 2 (\textup{w})\)

Now we are left with:

\(\displaystyle 46 = 2 (\textup{w})\)

We can divide both sides by 2 to find the width of the room.

\(\displaystyle \frac{46}{2} = \frac{2}{2} \textup{(w)}\)

\(\displaystyle \textup{w} = 23\)

Therefore, the lengths of the other two sides of the room are 23 ft.

To check to see if our answer is correct we can add the four sides of the room:

\(\displaystyle 15 + 15 + 23 + 23 = ?\)

This equals 76! (The perimeter of the room in the problem.)

Madison should buy a rug that is 15 ft. by 23 ft.

To find perimeter, simply use the formula for the perimeter of a rectangle.

\(\displaystyle P=2(w+l)=2(5+7)=2*12=24\)

To find perimeter of a rectangle, simply use the formula below:

\(\displaystyle P=2(w+l)=2(9+6)=2*15=30\)

\(\displaystyle P=2l+ 2w\)

We have the perimeter and the width, so we can plug those values into our equation and solve for our unknown. 

\(\displaystyle 42=2l+2(7)\)

\(\displaystyle 42=2l+14\)

Subtract \(\displaystyle 14\) from both sides

\(\displaystyle 42-14=2l+14-14\)

\(\displaystyle 28=2l\)

Divide \(\displaystyle 2\) by both sides

\(\displaystyle \frac{28}{2}=\frac{2l}{2}\)

\(\displaystyle 14=l\)

\(\displaystyle P=2l+ 2w\)

We have the perimeter and the width, so we can plug those values into our equation and solve for our unknown. 

\(\displaystyle 62=2l+2(8)\)

\(\displaystyle 62=2l+16\)

Subtract \(\displaystyle 16\) from both sides

\(\displaystyle 62-16=2l+16-16\)

\(\displaystyle 46=2l\)

Divide \(\displaystyle 2\) by both sides

\(\displaystyle \frac{46}{2}=\frac{2l}{2}\)

\(\displaystyle 23=l\)

\(\displaystyle P=2l+ 2w\)

We have the perimeter and the width, so we can plug those values into our equation and solve for our unknown. 

\(\displaystyle 92=2l+2(21)\)

\(\displaystyle 92=2l+42\)

Subtract \(\displaystyle 42\) from both sides

\(\displaystyle 92-42=2l+42-42\)

\(\displaystyle 50=2l\)

Divide \(\displaystyle 2\) by both sides

\(\displaystyle \frac{50}{2}=\frac{2l}{2}\)

\(\displaystyle 25=l\)