SOLUTION: use slopes to show that the quadrilateral whose vertices are (1.-1), (4,1),(2,2), and (5,4) is a parallelogram

Algebra ->  Geometry-proofs -> SOLUTION: use slopes to show that the quadrilateral whose vertices are (1.-1), (4,1),(2,2), and (5,4) is a parallelogram      Log On


   

Question 650170: use slopes to show that the quadrilateral whose vertices are (1.-1), (4,1),(2,2), and (5,4) is a parallelogram
Found 2 solutions by sachi, MathLover1:
Answer by sachi(548) About Me  (Show Source):
You can put this solution on YOUR website!
let the vertices of a quadrilateral are A(1.-1), B(4,1),C(2,2), and D(5,4)
then slope AB=[1-(-1)]/[4-1]=2/3
slope CD=[4-2]/[5-2]=2/3
so AB parallel to CD
now
slope AC=[2-(-1)]/[2-1]=3
slope BD=[4-1]/[5-4]=3
so AC parallel to BD
so ABCD is a parallelogram
ans

Answer by MathLover1(20850) About Me  (Show Source):
You can put this solution on YOUR website!
first, a slope for (1.-1), (4,1)
Solved by pluggable solver: Finding the slope


Slope of the line through the points (1, -1) and (4, 1)



m+=+%28y%5B2%5D+-+%28y%5B1%5D%29%29%2F%28x%5B2%5D+-+x%5B1%5D%29


m+=+%281+-+%28-1%29%29%2F%284+-+1%29


m+=+%281+%2B+1%29%2F%284+-+1%29


m+=+%282%29%2F%283%29


Answer: Slope is m+=+2%2F3




next, a slope for (2,2), and (5,4)
Solved by pluggable solver: Finding the slope


Slope of the line through the points (2, 2) and (5, 4)



m+=+%28y%5B2%5D+-+y%5B1%5D%29%2F%28x%5B2%5D+-+x%5B1%5D%29


m+=+%284+-+2%29%2F%285+-+2%29


m+=+%282%29%2F%283%29


Answer: Slope is m+=+2%2F3



lines have same slope; s0, they are parallel
next, a slope for (1.-1), and (2,2)
Solved by pluggable solver: Finding the slope


Slope of the line through the points (1, -1) and (2, 2)



m+=+%28y%5B2%5D+-+%28y%5B1%5D%29%29%2F%28x%5B2%5D+-+x%5B1%5D%29


m+=+%282+-+%28-1%29%29%2F%282+-+1%29


m+=+%282+%2B+1%29%2F%282+-+1%29


m+=+%283%29%2F%281%29


m+=+3



Answer: Slope is m+=+3



and, a slope for (1.-1), and (2,2)
Solved by pluggable solver: Finding the slope


Slope of the line through the points (4, 1) and (5, 4)



m+=+%28y%5B2%5D+-+y%5B1%5D%29%2F%28x%5B2%5D+-+x%5B1%5D%29


m+=+%284+-+1%29%2F%285+-+4%29


m+=+%283%29%2F%281%29


m+=+3



Answer: Slope is m+=+3



lines have same slope; s0, they are parallel
if both pairs of sides are parallel, then you can conclude that those four points define the vertex+points of a parallelogram


as you can see, their intersection is a parallelogram