Chapter Test
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In a parallelogram both pairs of opposite sides are parallel.
The diagonals of a parallelogram bisect each other.
(2,0)
(1,1)
In a parallelogram both pairs of opposite sides are parallel. Let's call the coordinates of vertex M (x,y), then use the Slope Formula to calculate the slope of each side of the parallelogram.
| Side | Slope Formula | Simplify |
|---|---|---|
| Slope of JK, ( - 2,- 1), ( 0,2) | 2-( -1)/0-( - 2) | 3/2 |
| Slope of KL, ( 0,2), (4,3) | 3- 2/4- 0 | 1/4 |
| Slope of LM, (4,3), (x,y) | y-3/x-4 | y-3/x-4 |
| Slope of MJ, (x,y), ( - 2,- 1) | -1 -y/- 2-x | -1-y/- 2-x |
Since the opposite sides in a parallelogram are parallel, we know the slopes of LM and MJ. Let's put these into the table.
| Side | Slope Formula | Simplify |
|---|---|---|
| Slope of JK, ( - 2,- 1), ( 0,2) | 2-( -1)/0-( - 2) | 3/2 |
| Slope of KL, ( 0,2), (4,3) | 3- 2/4- 0 | 1/4 |
| Slope of LM, (4,3), (x,y) | y-3/x-4 | y-3/x-4=3/2 |
| Slope of MJ, (x,y), ( - 2,- 1) | -1 -y/- 2-x | -1-y/- 2-x=1/4 |
(I): x= 2+4y
(II): y= 0
(II): Zero Property of Multiplication
(II): Identity Property of Addition
The intersection is the midpoint of each diagonal. Let's use the Midpoint Formula to find the midpoint of KM.
| Side | Points | (x_1+x_2/2,y_1+y_2/2) | Midpoint |
|---|---|---|---|
| KM | ( 0,2), ( 2,0) | (0+ 2/2,2+ 0/2) | (1,1) |
The coordinates of the intersection of the diagonals are (1,1).