SOLUTION: Hi, I have been struggling to this kind of problems can you help me? Here is the problem. Show in two ways that (-3,2), (-6,-2), (-1,-2) and (2,2) are the vertices of a rhombus.

Algebra ->  Length-and-distance -> SOLUTION: Hi, I have been struggling to this kind of problems can you help me? Here is the problem. Show in two ways that (-3,2), (-6,-2), (-1,-2) and (2,2) are the vertices of a rhombus.       Log On


   

Question 1197662: Hi, I have been struggling to this kind of problems can you help me? Here is the problem. Show in two ways that (-3,2), (-6,-2), (-1,-2) and (2,2) are the vertices of a rhombus.
Found 2 solutions by ewatrrr, math_tutor2020:
Answer by ewatrrr(24785) About Me  (Show Source):
You can put this solution on YOUR website!
2 hours, 30 minutes

Hi  
Show in two ways that (-3,2), (-6,-2), (-1,-2) and (2,2) are the vertices of a rhombus.
I.  Opposites Sides Parallel: 
Horizontal Sides Parallel and Slanted Sides, m= 4/3 for both 
II. Sides of Equal Lengths:
Use D = +sqrt+%28+%28x%5B1%5D-x%5B2%5D%29%5E2%2B+%28y%5B1%5D-y%5B2%5D%29%5E2+%29+:
 to show the sides are of Equal length:
For Example: AB = sqrt%283%5E2+%2B4%5E2%29 = 5




Wish You the Best in your Studies.

Answer by math_tutor2020(3817) About Me  (Show Source):
You can put this solution on YOUR website!

Method 1) Show that all four sides are the same length.

Let
A = (-3,2)
B = (-6,-2)
C = (-1,-2)
D = (2,2)

Use the distance formula to find the distance from A to B; i.e. the length of segment AB.
(x1,y1) = (-3,2) and (x2,y2) = (-6,-2)
d+=+sqrt%28+%28x1-x2%29%5E2+%2B+%28y1-y2%29%5E2+%29

d+=+sqrt%28+%28-3-%28-6%29%29%5E2+%2B+%282-%28-2%29%29%5E2+%29

d+=+sqrt%28+%28-3%2B6%29%5E2+%2B+%282%2B2%29%5E2+%29

d+=+sqrt%28+%283%29%5E2+%2B+%284%29%5E2+%29

d+=+sqrt%28+9+%2B+16+%29

d+=+sqrt%28+25+%29

d+=+5
Side AB is exactly 5 units long.

Repeat this set of steps to find the lengths of these three other sides:
BC
CD
AD
You should get 5 as the result of each segment if this figure is truly a rhombus.

==============================================================================

Method 2) Perpendicular diagonals and parallel sides.

The diagonals are AC and BD assuming the vertices are arranged in the order ABCD (clockwise or counterclockwise).

Let's find the slope of AC
m+=+%28y%5B2%5D+-+y%5B1%5D%29%2F%28x%5B2%5D+-+x%5B1%5D%29

m+=+%28-2+-+2%29%2F%28-1+-+%28-3%29%29

m+=+%28-2+-+2%29%2F%28-1+%2B+3%29

m+=+%28-4%29%2F%282%29

m+=+-2
Line AC has a slope of -2.

Now find the slope of diagonal BD
m+=+%28y%5B2%5D+-+y%5B1%5D%29%2F%28x%5B2%5D+-+x%5B1%5D%29

m+=+%282+-+%28-2%29%29%2F%282+-+%28-6%29%29

m+=+%282+%2B+2%29%2F%282+%2B+6%29

m+=+%284%29%2F%288%29

m+=+1%2F2
The slope of line BD is 1/2.

The two slopes (-2 and 1/2) multiply to -1 which shows the diagonals are perpendicular. Put another way, each slope is the negative reciprocal of one another.

Unfortunately the diagonals being perpendicular is not enough information to conclude we have a rhombus. The figure might be a kite.

The last thing we need is to show that the opposite sides AB and CD are parallel; as well as BC being parallel to AD.

Use the slope formula to find the slope of side AB
m+=+%28y%5B2%5D+-+y%5B1%5D%29%2F%28x%5B2%5D+-+x%5B1%5D%29

m+=+%28-2+-+2%29%2F%28-6+-+%28-3%29%29

m+=+%28-2+-+2%29%2F%28-6+%2B+3%29

m+=+%28-4%29%2F%28-3%29

m+=+4%2F3

Now do the same for side CD.
m+=+%28y%5B2%5D+-+y%5B1%5D%29%2F%28x%5B2%5D+-+x%5B1%5D%29

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

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

m+=+4%2F3
We get the same result which shows that AB is parallel to CD.

I'll let you check to see if BC is parallel to AD. That's the last piece of the puzzle to complete method 2, in showing we have a rhombus.