One more problem on Geometry...

Prakash Panneer · 2006-10-07 02:28:11

Given P,Q,R and S are the respective midpoints of sides AB , BC,CD , and AD of Quadrilateral
ABCD . Prove Quadrilateral PQRS is parallelogram.

Thanks in advance

sk · 2006-10-07 03:20:24

Hi!

Wait! I am working on it.

Let me reply within a few seconds.

Good question.

Prakash Panneer · 2006-10-07 07:58:15

Please help me...

luca-deltodesco · 2006-10-07 09:16:41

one proof of a parallelogram, is if the midpoints of the diagonals of said quadrilateral are the same.

and i dunno lol. but that could help maybe.

Toast · 2006-10-08 22:40:08

Huh?

To quote wikipedia, "All parallelograms are quadrilaterals"

They both adhere to the rules,
1. Opposite sides parallel
2. Angles bisection cross each other

luca-deltodesco · 2006-10-09 00:56:34

all parallelograms are quadrilaterals
but not all quadrilaterals are parallelograms

a parallelogram is a special case of a quadrilateral where opposite angles are equal, and opposite sides are equal, which it just so happens, means that midpoint of each diagonal are the same.

a rectangle is a special case of a parallelogram, where all angles are the same.
and a square is a special case of a rectangle, where all sides are the same.

all squares are rectangles
not all rectangles are squares

all squares are rectangles, of which all are parallelograms
but not all parallelograms are rectangles, nor are they all squares

all parallelograms are quadrilaterals,
but not all quadrilaterals are parallelograms

for example,

every number in the 12x table, is an even number
but not all even numbers are in the 12x table.

Devantè · 2006-10-09 00:58:13

That reads just like poetry. It has rhythm. Weird.

Last edited by Devanté (2006-10-09 00:58:25)

luca-deltodesco · 2006-10-09 01:13:51

Devanté wrote:

That reads just like poetry. It has rhythm. Weird.

lol. i guess it does kind of.

Devantè · 2006-10-09 01:15:22

Maybe because you have the facts bunched together to look like couplets...

Dross · 2006-10-09 02:38:32

Prakash Panneer wrote:

Given P,Q,R and S are the respective midpoints of sides AB , BC,CD , and AD of Quadrilateral
ABCD . Prove Quadrilateral PQRS is parallelogram.
Thanks in advance:up

Have a look at the images below. The first is any random quadrilateral.

To get the second, take afforementioned random quadrilateral and rotate it so that two opposite corners are horizontal (this is purely to make it easier to see what's going on).

Next, consider the triangle made with the top two parts of the quadrilateral, and the horizontal line at their base. When we draw the line between the mid-points of the top two parts of the quadrilateral, we see that it must be parallel to the horizontal line we've drawn. No matter how those two top parts are drawn, they'll always finish the same vertical distance away from the horizontal line (sice they must, after all, meet at their ends) and so their mid-points will always be the same distance away from said vertical line.*

Now, since this also applies to the other triangle (using the same horizontal line from above with the two lower sides of the quadrilateral), both these lines are parallel to each other.

Also, this applies if we rotate the quadrilateral so the other two corners are on the horizontal.

This shows that if you connect the mid-points of any quadrilateral, the new quadrilateral will have the property that each side is parallel to the opposite side - a parallelogram.

* - This bit is fairly important - Is it clear?

Last edited by Dross (2006-10-09 02:40:10)

Math Is Fun Forum

#1 2006-10-07 02:28:11

One more problem on Geometry...

#2 2006-10-07 03:20:24

Re: One more problem on Geometry...

#3 2006-10-07 07:58:15

Re: One more problem on Geometry...

#4 2006-10-07 09:16:41

Re: One more problem on Geometry...

#5 2006-10-08 22:40:08

Re: One more problem on Geometry...

#6 2006-10-09 00:56:34

Re: One more problem on Geometry...

#7 2006-10-09 00:58:13

Re: One more problem on Geometry...

#8 2006-10-09 01:13:51

Re: One more problem on Geometry...

#9 2006-10-09 01:15:22

Re: One more problem on Geometry...

#10 2006-10-09 02:38:32

Re: One more problem on Geometry...

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