The vertices of the square are the coordinates of the endpoints
The third vertex of the square could be (0,7) or (-5,1)
<h3>How to determine the third vertex</h3>
Two vertices of the square are:
(0,-3) and (4,2)
Calculate the distance between these vertices using:
![d = \sqrt{(x_2 - x_1)^2+(y_2 - y_1)^2}](https://tex.z-dn.net/?f=d%20%3D%20%5Csqrt%7B%28x_2%20-%20x_1%29%5E2%2B%28y_2%20-%20y_1%29%5E2%7D)
Square both sides
![d^2 = (x_2 - x_1)^2+(y_2 - y_1)^2](https://tex.z-dn.net/?f=d%5E2%20%3D%20%28x_2%20-%20x_1%29%5E2%2B%28y_2%20-%20y_1%29%5E2)
The above expression represents the area of the square.
So, we have:
![(x_2 - x_1)^2+(y_2 - y_1)^2 = 41](https://tex.z-dn.net/?f=%28x_2%20-%20x_1%29%5E2%2B%28y_2%20-%20y_1%29%5E2%20%3D%2041)
Next, we test the options with the given vertices (0,-3) and (4,2)
So, we have:
<u>A. (0,7)</u>
![(0 - 0)^2+(7 + 3)^2 = 41](https://tex.z-dn.net/?f=%280%20-%200%29%5E2%2B%287%20%2B%203%29%5E2%20%3D%2041)
![100 \ne 41](https://tex.z-dn.net/?f=100%20%5Cne%2041)
![(0 - 4)^2+(7 - 2)^2 = 41](https://tex.z-dn.net/?f=%280%20-%204%29%5E2%2B%287%20-%202%29%5E2%20%3D%2041)
![41 = 41](https://tex.z-dn.net/?f=41%20%3D%2041)
<u>B. (-5,1)</u>
![(-5 - 0)^2+(1 + 3)^2 = 41](https://tex.z-dn.net/?f=%28-5%20-%200%29%5E2%2B%281%20%2B%203%29%5E2%20%3D%2041)
![41 = 41](https://tex.z-dn.net/?f=41%20%3D%2041)
![(-5 - 4)^2+(1 -2)^2 = 41](https://tex.z-dn.net/?f=%28-5%20-%204%29%5E2%2B%281%20-2%29%5E2%20%3D%2041)
![82 \ne 41](https://tex.z-dn.net/?f=82%20%5Cne%2041)
<u>C.(5,13)</u>
![(5 - 0)^2+(13 + 3)^2 = 41](https://tex.z-dn.net/?f=%285%20-%200%29%5E2%2B%2813%20%2B%203%29%5E2%20%3D%2041)
![281 \ne 41](https://tex.z-dn.net/?f=281%20%5Cne%2041)
![(5 - 4)^2+(13 -2)^2 = 41](https://tex.z-dn.net/?f=%285%20-%204%29%5E2%2B%2813%20-2%29%5E2%20%3D%2041)
![122 \ne 41](https://tex.z-dn.net/?f=122%20%5Cne%2041)
<u>D.(-2,-2)</u>
![(-2 - 0)^2+(-2 + 3)^2 = 41](https://tex.z-dn.net/?f=%28-2%20-%200%29%5E2%2B%28-2%20%2B%203%29%5E2%20%3D%2041)
![5 \ne 41](https://tex.z-dn.net/?f=5%20%5Cne%2041)
![(-2 - 4)^2+(-2 -2)^2 = 41](https://tex.z-dn.net/?f=%28-2%20-%204%29%5E2%2B%28-2%20-2%29%5E2%20%3D%2041)
![52 \ne 41](https://tex.z-dn.net/?f=52%20%5Cne%2041)
<u>E.(4,-6)</u>
![(4 - 0)^2+(-6 + 3)^2 = 41](https://tex.z-dn.net/?f=%284%20-%200%29%5E2%2B%28-6%20%2B%203%29%5E2%20%3D%2041)
![25 \ne 41](https://tex.z-dn.net/?f=25%20%5Cne%2041)
![(4 - 4)^2+(-6 -2)^2 = 41](https://tex.z-dn.net/?f=%284%20-%204%29%5E2%2B%28-6%20-2%29%5E2%20%3D%2041)
![64 \ne 41](https://tex.z-dn.net/?f=64%20%5Cne%2041)
The vertices where at least one of the equations is true could be a third vertex of the square
Hence, a third vertex of the square could be (0,7) or (-5,1)
Read more about square vertices at:
brainly.com/question/1292795