Answer:
38 units
Step-by-step explanation:
You want the perimeter of pentagon ABCDE with vertices at A(4, -2), B(4, 10), C(7, 10), D(15, 4), E(7, -2).
Perimeter
The perimeter of a plane figure is the sum of the lengths of its sides.
Vertical, Horizontal sides
By counting grid squares, or subtracting coordinates, you can find the lengths of the vertical and horizontal segments easily:
AB = 12, BC = 3, EA = 3
Diagonal sides
The lengths of the diagonal segments can be found using the distance formula or your knowledge of Pythagorean triples. Counting grid squares (or subtracting coordinates), you see that each diagonal side has a vertical distance of 6 and a horizontal distance of 8. These correspond to twice the lengths of the sides in a 3-4-5 triangle, so you know the diagonal length is twice the hypotenuse: 2·5 = 10.
The Pythagorean theorem, or the distance formula, will tell you the same thing:
c² = a² +b² . . . . . . Pythagorean theorem for sides a, b, hypotenuse c
c = √(a² +b²)
diagonal length = √(6² +8²) = √100 = 10
Then the remaining two sides are ...
CD = 10, DE = 10
Application
The perimeter is ...
perimeter = AB +BC +CD +DE +EA
= 12 +3 +10 +10 +3 = 38 . . . units
The perimeter of pentagon ABCDE is 38 units.
