Take a quadrilateral and connect its 4 midpoints to make a new quad. what does the old quad have to be to make the new quad a rectangle?
The quadrilateral formed by joining the midpoint of consecutive sides of any quadrilateral is a parallelogram. Thus it is sufficient to show that two adjacent of the new quadrilateral are perpendicular. But adjacent of the new quadrilateral are each parallel to one of the diagonals of the original quadrilateral.
What sort of convex quadrilateral has perpendicular diagonals?
Also, note that the quadrilateral need not be convex.
Hello, stones44!
We're getting mixed messages now . . .
Connecting consecutive midpoints of any quadrilateral produces a parallelogram.Take a quadrilateral and connect its 4 midpoints to make a new quad.
What does the old quad have to be to make the new quad a rectangle?
Proof
Sketch any quadrilateral ABCD.
Let D, E, F, G be the midpoints of AB, BC, CD, DA, respectively.
Draw diagonal AC; draw segments DE and FG.
In ∆ABC, DE is parallel to AC and DE = ½·AC.
In ∆ADC, FG is parallel to AC and FG = ½·AC.
. . The segment joining the midpoints of two sides of a triangle
. . is parallel to and one-half the length of the third side.
Therefore, DEFG is a parallelogram.
. . If two sides of a quadrilateral are parallel and equal,
. . the quadrilateral is a parallelogram.
If the new quad is to be a rectangle,
. . the original quad must have perpendicular diagonals.
There is no specific name for this type of figure.
Take a look . . .Code:* * | * * | * * | * * - - - + - - - - - - - - - - - * * | * * | * * | * * | * * | * * | * *| * *
Why is that the case?
In the case of a convex quadrilateral, if its diagonals are perpendicular then the mid-point quadrilateral is a rectangle. So again I ask you “What sort of convex quadrilateral has perpendicular diagonals”?
I gave you a example of a non-convex quadrilateral in which the mid-point quadrilateral is a rectangle. However, I think that whoever asked the question meant convex.
If you are confused, don’t be. The first response you received is incorrect.
soroban...you helped a lot ---thanks
but how do you show that the diagonals must be perpendicular...if you have 4 quads (that make up the //ogram when the diagonals cut them up), and each has one 90 angle, that doesnt show too much about the diagonals being 90