PERIMETER 

The distance around a geometric figure is called the perimeter. You can use what you have learned about combining like terms to simplify expressions for perimeter.

A closed figure with straight sides is called a polygon. To find the perimeter of a polygon, add the lengths of the sides.

Example 1 : 

Write an expression for the perimeter of the quadrilateral shown below. 

Solution : 

Add the lengths of the four sides.

P  =  (a + 3) + (2a - 8) + (3a - 3) + (a - 1)

Combine like terms to simplify.

=  (a + 2a + 3a + a) + (3 - 8 - 3 - 1)

=  7a - 9

Example 2 : 

Find the perimeter of the quadrilateral above in example 1 for a = 5.

Solution :

Substitute 5 for a in (7a - 9).

P  =  7(5) - 9

Multiply; then subtract.

=  35 - 9

=  26 units

Example 3 : 

What happens to the perimeter of the triangle shown below when the length of each side is doubled?

Solution :

Write an expression for the perimeter of the triangle shown above.

P  =  (x - 1) + (2x - 1) + (x + 3)

Combine like terms to simplify the expression.

=  (x + 2x + x) + (-1 - 1 + 3)

=  4x + 1

Double the length of each side of the triangle.

2 (x - 1)  =  2x - 2

2 (2x - 1)  =  4x - 2

2 (x + 3)  =  2x + 6

Find the perimeter for the new side lengths. 

=  (2x - 2) + (4x - 2) + (2x + 6)

Combine like terms to simplify.

=  (2x + 4x + 2x) + (-2 - 2 + 6)

=  8x + 2

Use the Distributive Property. 

=  2(4x + 1)

=  2 ⋅ Perimeter of the original triangle

When the length of each side of the triangle is doubled, perimeter is also doubled. 

Example 4 : 

What happens to the perimeter of this trapezoid when the length of each side is tripled? Use the variables a, b, b, and c for the lengths of the sides.

Solution :

Write an expression for the perimeter of the trapezoid shown above.

P  =  a + b + b + c

=  a + 2b + c

Triple the length of each side.

3a, 6b, 3c

Find the perimeter of the new side lengths. 

=  3a + 6b + 3c

Use the Distributive Property.

=  3(a + 2b + c)

=  3 ⋅ Perimeter of the original trapezoid

When the length of each side of the trapezoid is tripled, perimeter is also tripled. 

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