math question can u solve?

Question

given a -2,1,b2,5,c6,-1 and d4,-7 form a quarrilateral p q r and s are the midpoints of the sides ab, bc, cd, and da respectively . prove that pqrs is a parallelogram

Answer 1

this is a fairly easy question but requires a few steps..
ill try & explain it as best i can
here goes:

first, plot your points, a, b, c, and d
then even though you can probably see where the midpoints would go, im pretty sure you have to prove it with the midpoint formula, which is like this:
[(x_1 + x_2)/2 , (y_1 + y_2)/2]
when you plug the numbers in, you end up with a new set of coordinates, which are your midpoint.
so, for line AB:
[(-2+2)/2, (1+5)/2]
(0/2, 6/2), and you get:
P (0, 3)

Do this formula with each line. The coordinates for your midpoints would end up being:
P (0, 3)
Q (4, 2)
R (5, -4)
S (1, -3)

Then connect these points and you end up with parallelogram PQRS

So now you have to prove that it is a parallelogram
Do this by using the slope formula (change in y/change in x) or (rise/run)

The slope of PQ is -1/4 because you go down one and right 4
The slope of SR is also -1/4 for the same reason. This proves that these two lines are parallel because they have the same slope.

Now for the other two lines:
The slope of SP is -6/1--you go down 6 and right one to get from P to S
The same goes for line QR, you go over down 6 and right 1, making the slope -6/1
Therefore, those two lines are parallel and you have just proven that it is a parallelogram!

Hope I helped and wasnt too confusing =]

Answer 2

A (-2, 1) B (2, 5) C (6, -1) and D (4, -7)

P = Midpoint AB = ((-2 + 2)/2, (1 + 5)/2) = (0, 3)
Q = Midpoint BC = ((2 + 6)/2, (5 - 1)/2) = (4, 2)
R = Midpoint CD = ((6 + 4)/2, (-1 - 7)/2) = (5, -4)
S = Midpoint DA = ((4 - 2)/2, (-7 + 1)/2) = (1, -3)

Slope of PQ = (2 - 3)/(4 - 0) = - 1/4
Slope of QR = (-4 - 2)/(5 - 4) = - 6
Slope of RS = (-3 + 4)/(1 - 5) = - 1/4
Slope of SP = (3 + 3)/(0 - 1) = - 6

Because of equal slopes
PQ is parallel to RS and
QR is parallel to SP

So PQRS is a parallelogram

Answer 3

Is this supposed to be read as "a" is the point (-2, 1), "b" is (2,5), "c" is (6, -1), and "d" is (4, -7)?

Use the midpoint formula to find the coordinates of p, q, r, and s. Show that the slopes between consective points are only one of two different slopes, and that opposite sides have the same slope. This proves that pqrs is a parallelogram.

Answer 4

given a(-2,1), b(2,5), c(6,-1) and d(4,-7),
midpoint of ab is p(0,3), mid of bc is q(4,2), mid of cd is r(5,-4), mid of ac is s(1,-3)
slope of pq = -1/4 = slope of rs, and
slope of sp = -6 = slope of rq,
so opposite sides are parallel, which by definition is a parallelogram.

Answer 5

p = midpoint of ab = (0,3)
q = midpoint of bc = (4,2)
r = midpoint of cd = (5,-4)
s = midpoint of da = (1, -3)
Now show that slope of pq =slope rs
and slope qr = slope sp.

That shows opposite sides are || and thus you have a ||-ogram