SOLUTION: prove by three methods that the points (5,3), (-8,-5), (-6,-6) and (7,2) form a parallelogram??

Algebra.Com
Question 1040566: prove by three methods that the points (5,3), (-8,-5), (-6,-6) and (7,2) form a parallelogram??
Found 2 solutions by Edwin McCravy, Boreal:
Answer by Edwin McCravy(20081)   (Show Source): You can put this solution on YOUR website!


There are four methods to show it:

1.  A parallelogram is a quadrilateral with both pairs of 
opposite sides parallel. 

A. Use the slope formula to show that AB is parallel to CD
B. Use the slope formula to show that BC is parallel to AD


2,  If both pairs of opposite sides of a quadrilateral are 
congruent, the quadrilateral is a parallelogram. 

A. Use the distance formula to show that AB and CD have the 
same length and therefore are congruent.
B. Use the distance formula to show that BC and AD have the 
same length and therefore are congruent.

3.  If the diagonals of a quadrilateral bisect each other, 
the quadrilateral is a parallelogram. 

A. Use the midpoint formula to find the midpoint of diagonal AC.
B. Use the midpoint formula to find the midpoint of diagonal BD 
to show that the two midpoints are the same.

4.  If ONE PAIR of opposite sides of a quadrilateral are BOTH 
parallel and congruent, the quadrilateral is a parallelogram.

If you have done both 1 and 2, you have already done enough to 
show that AD and BC are both parallel and congruent.

If you have trouble, tell me in the thank-you note form below and
I'll get back to you by email.

Edwin
Answer by Boreal(15235)   (Show Source): You can put this solution on YOUR website!
Draw it. The lines between the two Quadrant III points and the other two points should have the same slope and be parallel.
The slope is -1/2 between (-8,-5) and (-6,-6). It is -1/2 for the other two points, too.
Between (-6,-6) and (7.2) slope is 8/13. That should be the same between (5,3) and (-8.-5). It is.
The sides should be equal length. The distance between the third quadrant points is sqrt(1^2+2^2) The distance between the two first quadrant points is (1^2+2^2). For the longer sides, (-6,-6) and (7,2), it is sqrt (169+64), and for the other two (5,3) and (-8,-5), it is sqrt (64+169).
For the angles, one may show that the diagonals intersect at their midpoint. Then the triangles formed would be congruent on the basis of SAS (the angles being vertical angles).
Diagonal from (-6,-6) to (5,3) has slope 9/11 and its equation is y-3=(9/11)(x-5)
That is y=(9/11)x-(12/11)
The other diagonal is from (-8,-5) to (7,2). Its slope is (7/15) and function is y-2=(7/15)(x-7). That is
y=(7/15)x-19/15
The intersection of the two diagonals is solving this system.
11y-9x=-12
15y-7x=-19
77y-63x=-84
-135y+63x=+171
-58y=87; y=-3/2
-16.5-9x=-12
-9x=4.5; x=-0.5
Check into second. -22.5+3.5=-19
intersect at (-0.5,-1.5)
The midpoint of the first diagonal is (-.5,-1.5)
The midpoint of the second diagonal is (-.5,-1.5)
The angles between the half diagonals are equal.
SAS and the angles are equal.
RELATED QUESTIONS

Okay I have to graph the points A(-5,0) , B(3, 2) C(5, 6) and D(-3 4). Then show that... (answered by rothauserc)
The points (-7, -11) (11, 5) (6, 11) and (-12,-5) are vertices of a parallelogram. Draw... (answered by Alan3354)
Calculate the area of the triangle with the following vertices: 6. (-3, -3), (0, -3), (answered by Alan3354)
show that the vertices represented by the complex numbers 6-i, 7+3i, 8+2i & 7-2i form a... (answered by ikleyn)
Three of the vertices of a parallelogram are (2, 1) (4, 7), and (6, 5). What are the... (answered by KMST)
Show that the points (-1, -3) (-2, 0) (1, 6) and (2, 3) are the vertices of a... (answered by Boreal)
Given the points (1,4), (3,6), (1,0) and (3,2), graph to prove it is a parallelogram.... (answered by greenestamps)
Use the following ordered pairs: (-5,5), (-5,-8), (8,5) 1. Calculate each side of the... (answered by Alan3354)
Prove that the points A(-2,-3), B(6,2), C(8,7) and D(0,2) are the vertices of a... (answered by math_helper)