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.

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) (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

-------------------------

Start with the slope formula

Plug in ,,,

Subtract the terms in the numerator to get . Subtract the terms in the denominator to get

Reduce

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

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

-------------------------

Start with the slope formula

Plug in ,,,

Subtract the terms in the numerator to get . Subtract the terms in the denominator to get

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

Since the product of the two slopes is -1 (ie ), 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.
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