Answer:
18.8 units
Explanation:
You want the perimeter of the polygon defined by vertices (-2, -3), (-2, 1), (0, 3), (3, -1), and (3, -3).
Lengths
The attached figure plots the points and shows the distances between them. The lengths of sides aligned with the grid can be found by counting grid squares or finding the difference of coordinates.
Side BC is the diagonal of a square 2 units on a side, so has length 2√2, about 2.8.
Side CD is the hypotenuse of a 3-4-5 triangle, so is 5 units.
Perimeter
The perimeter is the sum of the side lengths of the figure:
P = AB +BC +CD +DE +EA
P = 4 +2.8 +5 +2 +5 = 18.8 . . . units
The perimeter of the polygon is about 18.8 units.
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Additional comment
Using the Pythagorean theorem, we can find BC as the hypotenuse of a right triangle with sides of length 2: √(2²+2²) = 2√(1+1) = 2√2. Similarly, side CD is the hypotenuse of a triangle with sides 3 and 4 units: √(3²+4²) = √(9+16) = √25 = 5.
It is useful to remember the ratios of side lengths of an isosceles right triangle: 1 : 1 : √2. Similarly, it is useful to remember a few of the common Pythagorean triples: {3, 4, 5}, {5, 12, 13}, {7, 24, 25}, {8, 15, 17}. These are the side lengths of right triangles with integer lengths. These and their multiples are often seen in problems like this and in other algebra, trig, and geometry problems.
In general, the distance formula is ...
d = √((x2 -x1)² +(y2 -y1)²)
You don't need to do this math if you can recognize distances without it.
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