Question

1. Write a coordinate proof for the following theorem below. Include the given and prove statements.
If a quadrilateral is a square, then its diagonals are congruent.
Square ABCD and its diagonal have been drawn for you.

2. The coordinates of 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

@whpalmer4

Answer 2

3. If you were to place a rhombus in the coordinate plane so that its diagonals lie along the axes, what would be the coordinate of the vertices using a few variables as possible?

Answer 3

I don't have to draw it out, I just have to show the steps to solve it and find the correct answer.

Answer 4

that's what we've got given to us for the first one, right?

Answer 5

we can pick our own coordinates so that the sides are parallel to the axes

Answer 6

This is for # 1 (sorry forgot to put these up.)

Answer 7

we know that the sides of a square are equal, right? so the diagonal AC we could find with Pythagoras given the lengths of AD and DC (or AB and BC) and similarly we could find BD with the lengths of DC and BC or AB and AD)

Answer 8

This is for #2!

Number three don't have a pic.

Answer 9

and as the drawing conveniently puts the square right on the axes, we can assign coordinates to those points. D is at (0,0), C is at (C,0), A is at (0,A), B is at (C,B)

Answer 10

but all the sides are the same, so we could just call the non-zero coordinate value \(s\) (for side length, I suppose) making the coordinates be (0,0), (s,0), (0,s), and (s,s)

again, easy to find the length of the diagonals with Pythagoras.