If you plot these points on a piece of graph paper, you will see that the point (-2,7) is directly above the point (-2,1) because they share the same x value. They both fall on a perfectly vertical line. If they both fall on a perfectly vertical line, all you have to do to find the length of that segment is to count the number of units 1 is above the other. That's 6. The same goes for the points (4,7) and (4,1). They are also on the same perfectly vertical line so they are also 6 units apart and the length of that segment is 6. Btw, this polygon looks like a house that a kid would draw with a pointy roof. The side lengths I just gave you were the sides of the house. In order to find the length of the base of the house, we count the x units between the 2 points (-2,1) and (4,1). The distance between -2 and 4 is 6. So far we have 3 measurements for this house, and all 3 are 6. Now we come to the pointy roof on the house. It's a line that is a diagonal, so we will use the distance formula here to get the correct measure. The distance formula is [tex]d= \sqrt{( x_{2}- x_{1} )^2+( y_{2}- y_{1})^2 } [/tex]. Filling in using the endpoints for that segment we have [tex]d= \sqrt{(1-(-2))^2+(11-7)^2} [/tex]. That simplifies to [tex]d= \sqrt{3^2+4^2} [/tex] which simplifies even further to [tex]d= \sqrt{25} [/tex] and d = 5. The other slant now...[tex]d= \sqrt{(1-4)^2+(11-7)^2} [/tex] which simplifies to [tex]d= \sqrt{(-3)^2+4^2} [/tex] which simplifies further to [tex]d= \sqrt{25} [/tex] and d = 5. So the side lengths for our polygon are 6+6+6+5+5 which equals 28 units. There you go!
