Shawn connected the four points (-4,1) (-1,4) (5,1) and (-1,-5) on a coordinate plane. Use slope to determine what type of quadrilateral was drawn

Question

You HAVE to use the slope to figure out the answer and show the work of the slopes as well

Answer 1

(-4,1), (-1,4), (5,1), (-1,-5)

(-4,1), (-1,4) slope=(4-1)/(-1-[-4])=3/3=1

(-1,4), (5,1) slope=1-4/(5-[-1])=-3/6=-1/2

(5,1), (-1,-5) slope=(-5-1)/-1-5)=-6/-6=1

(-1,-5), (-4,1) slope=(1-[-5])/-4-[-1])=6/-3=-2

the quadrilateral is a trapezoid because two lines have the same slope and therefore are parallel

also, using the distance formula, you get √(9+36)=√45=3√5 for (-1,4) and (5,1)

for (-4,1) and (-1,-5) you get √(9+36)=√45=3√5 which means that because the two non-parallel sides are equal, you have an isosceles trapezoid as well

Answer 2

Alright, well, it looks like they don't want any of those edges to intersect, so the edges of the quadrilateral will be between (-4,1) and (-1,4), (-1,4) and (5,1), (5,1) and (-1,-5), and (-1,-5) and (-4,1). So you just find the slope between each pair of points using our handy-dandy slope formula: (y1-y2)/(x1-x2), and you will find the find the  respective slopes 1, -1/2, 1, and -2. 

 

Now, what do these slopes tell us? First off, we know we have two 1's which let's us know that there are two sides that are parallel, so they must be opposite sides. The other two slopes aren't equal, which tells us they aren't parallel and that the two other, parallel, sides must be of different lengths. In the end you are stuck with a quadrilateral that is interesting only in that it has two parallel and different length sides, which is the precise definition of a trapezoid that doesn't fall into any other categories of quadrilaterals. So there it is, you have a trapezoid. 

 

NOTE: if you were confused about anything, here are a couple things to note: two lines are parallel if they have the same slope, yet perpendicular if they have opposite reciprocal slopes, and a trapezoid is defined as a quadrilateral with a pair of opposite parallel sides and is thus one of the broadest categories of quadrilateral.