SOLUTION: 2. Prove that the points (-2, 9), (-4, -2), (1, -12) and (3, -1) are vertices of a rhombus (a parallelogram with all sides equal length). Show that the diagonals are perpendicular.

Algebra ->  Graphs -> SOLUTION: 2. Prove that the points (-2, 9), (-4, -2), (1, -12) and (3, -1) are vertices of a rhombus (a parallelogram with all sides equal length). Show that the diagonals are perpendicular.      Log On


   

Question 127642: 2. Prove that the points (-2, 9), (-4, -2), (1, -12) and (3, -1) are vertices of a rhombus (a parallelogram with all sides equal length). Show that the diagonals are perpendicular.
Answer by jim_thompson5910(35256) About Me  (Show Source):
You can put this solution on YOUR website!


First let's find the slope of the line through the points (-2,9) and (1,-12) (these vertices are opposite from one other)




Let's denote the first point (-2,9) as . In other words, and

Now let's denote the second point (1,-12) as . In other words, and



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m=%28y%5B2%5D-y%5B1%5D%29%2F%28x%5B2%5D-x%5B1%5D%29 Start with the slope formula

m=%28-12-9%29%2F%281--2%29 Plug in y%5B2%5D=-12,y%5B1%5D=9,x%5B2%5D=1,x%5B1%5D=-2


m=-21%2F3 Subtract the terms in the numerator -12-9 to get -21. Subtract the terms in the denominator 1--2 to get 3

m=-7 Reduce


So the slope of the line through the points (-2,9) and (1,-12) is m=-7






Now let's find the slope of the line through the points (-4,-2) and (3,-1) (these vertices are opposite from one other)





Let's denote the first point (-4,-2) as . In other words, and

Now let's denote the second point (3,-1) as . In other words, and



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m=%28y%5B2%5D-y%5B1%5D%29%2F%28x%5B2%5D-x%5B1%5D%29 Start with the slope formula

m=%28-1--2%29%2F%283--4%29 Plug in y%5B2%5D=-1,y%5B1%5D=-2,x%5B2%5D=3,x%5B1%5D=-4


m=1%2F7 Subtract the terms in the numerator -1--2 to get 1. Subtract the terms in the denominator 3--4 to get 7



So the slope of the line through the points (-4,-2) and (3,-1) is m=1%2F7




Since the product of the two slopes is -1 (ie -7%281%2F7%29=-7%2F7=-1), this shows us that the two slopes are perpendicular. So the two diagonals are also perpendicular. This shows us that the figure is a rhombus.