1. ## graphing a square.

hi peoples of math help forum! -waves-

i hope this belongs in the geometry board, as my teacher said there are multiple ways to solve it. this is the question that has be stumped:

Points D and E are the endpoints of one of the sides of a square. If the coordinates of the midpoint of DE are (4.5, 16) and the coordinates of D are (2, 16), what are the coordinates of the vertices of the square.

thanks

2. Originally Posted by lyyy94
hi peoples of math help forum! -waves-

i hope this belongs in the geometry board, as my teacher said there are multiple ways to solve it. this is the question that has be stumped:

Points D and E are the endpoints of one of the sides of a square. If the coordinates of the midpoint of DE are (4.5, 16) and the coordinates of D are (2, 16), what are the coordinates of the vertices of the square.

thanks
(4.5-2)2=x where x is the length of an edge

3. i understand that the length is 5... i just dont understand how to find the points.

4. Originally Posted by lyyy94
hi peoples of math help forum! -waves-

i hope this belongs in the geometry board, as my teacher said there are multiple ways to solve it. this is the question that has be stumped:

Points D and E are the endpoints of one of the sides of a square. If the coordinates of the midpoint of DE are (4.5, 16) and the coordinates of D are (2, 16), what are the coordinates of the vertices of the square.

thanks
If the co-ordinates of E where (x, y), then

$\frac{x+2}{2} = \frac{9}{2}$

Find x. y remains the same.

So DE is a horizontal line. Find the length L of DE. If F and G are the other two vertices, then the co-ordinates of F will be, (x + or - L, 16 + or - L). Similarly you can find the co-ordinates of G.

5. Originally Posted by lyyy94
i understand that the length is 5... i just dont understand how to find the points.
The length of the square have equal sides. From the question, the endpoints can be anywhere. We could assume the line DE is the direct center of the square, DE is an exterior edge, DE is 1 unit up from the edge, etc. Depending on how we view the location of the line segment DE, our vertices can be almost anywhere.

6. I don't like how the question is worded. I can interrupt it to mean that the segment DE is an exterior edge or DE is just a line segment where D is on the left and E is the right.

I attached a photo of how I think the question could be interrupted. Where the floating DE line can be anywhere on the square.

7. I'm not fond of the wording either. I believe my teacher has interpreted it as being one of the edges.

8. Originally Posted by lyyy94
I'm not fond of the wording either. I believe my teacher has interpreted it as being one of the edges.
If it is an edge, then D and E are two vertices. The other two will be 5 units above or below D and E since again we don't know if this is a top or bottom of the square. Therefore, our coordinates will be of the form $(2,16\pm5)$ and $(7,16\pm5)$ depending on if you want DE to be the top or the bottom.

9. thank you for your help.

10. Hello, lyyy94!

Did you make a sketch?

Points $D$ and $E$ are the endpoints of one of the sides of a square.
If the coordinates of the midpoint of $DE$ are (4.5, 16) and the coordinates of $D$ are (2, 16),
what are the coordinates of the vertices of the square?
Code:
        |
|     D         M         E
|     * - - - - * - - - - *
|   (2,16)   (4½,16)    (7,16)
|
|
|
|
- - + - - - - - - - - - - - - - - - -
|

Since the midpoint $M$ is $2\tfrac{1}{2}$ units to the right of $D,$
. . $E$ must be $2\tfrac{1}{2}$ units to the right of $M.$

Hence, $E$ is at $(7,16)$
. . The side of the square is: $DE = 5.$

The other two vertices are either:

. . $\begin{array}{cccc}
\text{5 units above }D\text{ and }E\!: & (2,21),\;(7,21) \\
\text{or} \\
\text{5 units below }D\text{ and }E\!: & (2,11),\;(7,11) \end{array}$