Question

The coordinates for a rhombus are given as (2a,0), (0,2b), (-2a,0), and (0,-2b). Write a plan to prove that the midpoints of the sides of a rhombus determine a rectangle using coordinate geometry. Be sure to include the formulas

Answer 1

So you want the slopes of the intersecting line segments to be perpendicular, agreed?

Answer 2

Yeah

Answer 3

and I mean not the slopes to be perpendicular but the line segments to be perpendicular
i said that a little wrong
errr
but anyways the reason slopes came into mind is because we know what about the slopes if the line segments are perpendicular?

Answer 4

They are the opposite recipricols and I knew what you meant haha

Answer 5

and i guess they were talking about the midpoints

Answer 6

not where the line segments intersect

Answer 7

pretend that is perfect

Answer 8

they already give you the midpoints

Answer 9

just show the slopes are undefined and 0 with those points given

Answer 10

like look at one line segment at a time
Take the line segment with (-a,b) and (-a,-b)

Answer 11

find the slope

Answer 12

then do the same with the line segment going through (-a,b) and (a,b)

Answer 13

how do you find the slope of that though?

Answer 14

do you know the slope formula?
Also what do you know about the slopes of vertical lines and horizontal lines without using the slope formula?

Answer 15

yeah and the slope of horizontal lines is 0 and the slope of vertical lines is undefined but how do we know they are horizontal and vertical lines?

Answer 16

all the points on x=a have what in common?

Answer 17

look at the line segment going through the midpoints

Answer 18

oh I see now. nevermind that was a stupid question. Okay so now that I know the slopes of each line how do I prove its a rhombus?

Answer 19

you were trying to show that one thing is a rectangle

Answer 20

oops I meant to say how do I prove its a rectangle

Answer 21

we did that by showing the angles were 90 degrees.

Answer 22

We show that by showing the line segments were perpendicular.

Answer 23

we knew the line segments were perpendicular because one was horizontal and the other was vertical

Answer 24

we knew we had a horizontal line segment because all the y values of the coordinates that lied on that line segment were the same

Answer 25

Okay so there arent any formulas that I have to include or anything?

Answer 26

we knew we had a vertical line segment because all the x values of the coordinates that lied on that line segment were the same

Answer 27

If you want you could use that one formula I mentioned

Answer 28

The slope formula right? but I don't really need it

Answer 29

You also might want to use that midpoint formula to show why those points in your picture are midpoints.

Answer 30

Okay. that makes sense. Thanks for your help!

Answer 31

np