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# Hugo is writing a coordinate proof to show that the midpoints of a quadrilateral are the vertices of a parallelogram. He starts by assigning

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Hugo is writing a coordinate proof to show that the midpoints of a quadrilateral are the vertices of a parallelogram. He starts by assigning coordinates to the vertices of quadrilateral RSTVquadrilateral RSTV and labeling the midpoints of the sides of the quadrilateral as A, B, C, and D.

The coordinates of point A are (, ).

The coordinates of point B are (, ).

The coordinates of point C are (2c, d)(2c, d) .

The coordinates of point D are (c, 0)(c, 0) .

The slope of both AB¯¯¯¯¯AB¯ and DC¯¯¯¯¯DC¯ is .

Because both pairs of opposite sides are parallel, quadrilateral ABCDquadrilateral ABCD is a parallelogram.

?

viyiwells  Dec 29, 2017
edited by viyiwells  Dec 29, 2017
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So the coordinates of A and B are basically finding the mid-point of RS and ST.

1st  to find the mid-point u use the formula $$(\frac{x_1+x_2}{2}, \frac{y_1+y_2}{2})$$

The mid-point of RS is,

$$(\frac{0+2a}{2}, \frac{0+2b}{2})$$

$$(\frac{2a}{2}, \frac{2b}{2})$$

$$(a,b)$$

$$(a,b)$$ are the coordinates of A.

Now the midpoint of ST is,

$$(\frac{2a+2c}{2}, \frac{2b+2d}{2})$$

$$(\frac{2(a+c)}{2}, \frac{2(b+d)}{2})$$

$$((a+c),(b+d))$$

$$((a+c)+(b+d))$$ are the coordinates of ST.

I didn't quite understand what the slope part means

Rauhan  Dec 29, 2017
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For the slopes:

Use the slope formula:  m  =  (y2 - y1) / (x2 - x1)

Slope of AB:  [ (b + d) - b ] / [ (a + c) - a ]  =  d / c

Slope of DC: [ d - 0 ] / [ 2c - c ]  =  d / c

Since these slopes are equal, these sides are parallel.

Slope of AD:  =  ( b - 0 ) / ( a - c )  =  b / ( a - c )

Slope of BC  =  [ (b + d) - d ] / [ (a + c} - 2c ]  =  b / ( a - c )

Since these slopes are equal, these sides are parallel.

geno3141  Dec 29, 2017