We need to find an expression that represents the perimeter of each of the following figures. Then, we will simplify the found expression as much as possible. Recall that perimeter is the total length of all the sides added together.

a

We will use the length of a tile as our main unit. We can see that the length of the figure is

x + 1 + y + 1 + x,

while its height is just one tile. To get the perimeter, we will need to add

2

times the figure's length plus

2

times the figure's height. The perimeter is described by the following expression

P = 2 (x + 1 + y + 1 + x) + 2 (1) .

Now let's simplify the expression for the perimeter.

P = 2 (x + 1 + y + 1 + x) + 2 (1)

Distr

Distribute $2$

P = 2 x + 2 + 2 y + 2 + 2 x + 2

AddTerms

Add terms

P = 4 y + 4 x + 6

We have found the expression for the perimeter.

P = 4 y + 4 x + 6

b

We will use the length of one tile as our main unit. By labeling each side length we can visualize the perimeter.

Using the lengths above, we can write an expression for the perimeter.

P = x + 1 + (x - 1) + 1 + 1 + 1 + 1

Let's simplify the perimeter expression.

P = x + 1 + (x - 1) + 1 + 1 + 1 + 1

RemovePar

Remove parentheses

P = x + 1 + x - 1 + 1 + 1 + 1 + 1

AddSubTerms

Add and subtract terms

P = 2 x + 4

Therefore, the perimeter is given by the expression

P = 2 x + 4

c

We will use the length of one tile as our main unit. By labeling each side length we can visualize the perimeter.

Using the lengths above, we can write an expression for the perimeter.

P = y + 1 + 1 + 1 + (y - 2) + 1 + 1 + 1 + (x - 1) + 1 + 1 + 1 + y + x + y

Let's simplify the perimeter expression.

P = 4 y + 2 x + 6

d

We will use the length of one tile as our main unit. By labeling each side length we can visualize the perimeter.

Using the lengths above, we can write an expression for the perimeter.

P = y + 1 + 1 + 1 + (y - x - 1) + x + x + 1 + x + 1 + 1 + 1

Let's simplify the perimeter expression.

P = 2 y + 2 x + 6

Exercise