skidtheman 3 years ago Please Help Me with this 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. https://www.connexus.com/content/media/456845-212011-111754-AM-714814192.png

1. androidonyourface

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Let me finish my cookie and I'll get right on it. ;D

3. androidonyourface

omg thank you so much!

I was going to type out a proof on Word or something, but I decided that would be a waste of time. So here goes. Typed out in a chat-box. We know that the sides of right triangles scale isometrically as the triangle increases in size. That is to say, if the hypotenuse doubles, so do both sides. This can be demonstrated by trigonometry, i.e., tan(30)=3/4, sin(30)=3/5, but tan(30) could also be 6/8 and sin(30) could also be 6/10. So, this part is pretty self-evident. As a direct result, each of the rhombus' sides have their midpoints at half their respective components. So the line connecting (-2a,0) and (0,2b) is necessarily (-a,b), since half of (-2a,0)'s x component is -a, and half of (0,2b)'s y component is b. Repeat this for all sides to prove that the midpoints are the ones listed in the image. (-a,b), (a,b), (-a,-b), (a,-b). Now, let's call the line connecting (-a,b) and (a,b) line A. The length of line A is 2a, since a-(-a)=a+a=2a. It's also parallel to the line connecting (-a,-b) and (a,-b), which we'll call line B. We know they're parallel since the lines have slopes of m=0. We also know that the length of line B is 2a. Repeating this for the line connecting (-a,b) with (-a,-b) and (a,b) with (a,-b) we'll find that they're also parallel (with a slope of infinity) with the same length of 2b. Therefore, all four lines are connected, with opposite lines being parallel and of the same length. Hence, a rectangle.

5. androidonyourface

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