Answer:
The answer is A) All four sides are equal. The simplest radical form for each side is √40.
Explanation:
If X is plotted on 6 on the y-axis, then X is located at the point (2, 6), since W is plotted on 2 on the x-axis, W is located at the point (2, 2).
Since we know that XW is one side of the square, we can find the other three vertices of the square using the fact that all sides of a square are equal in length and perpendicular to each other.
To find the third vertex, we can use the fact that Z is plotted at (4, -2) and is perpendicular to XW. Since XW has a length of 4 (the difference in y-coordinates between X and W), we know that the length of ZY must also be 4. To find the coordinates of Y, we can move 4 units up from the y-coordinate of Z (which is -2), giving us a y-coordinate of 2. Since ZY is perpendicular to XW, we know that the x-coordinate of Y must be the same as the x-coordinate of Z (which is 4). Therefore, the coordinates of Y are (4, 2).
To find the fourth vertex, we can use the fact that all sides of the square are equal in length. Since XW has a length of 4, we know that ZY must also have a length of 4. Therefore, the fourth vertex must be located 4 units to the right of Y and 4 units up from X. This gives us a fourth vertex with coordinates of (6, 6).
Therefore, the vertices of the square are W(2, 2), X(2, 6), Y(4, 2), and Z(4, -2).
To check that the sides of the square are perpendicular to each other, we can calculate the slopes of the sides.
The slope of XW is:
m_XW = (6 - 2) / (2 - 2) = undefined
The slope of ZY is:
m_ZY = (2 - (-2)) / (4 - 4) = undefined
Since both slopes are undefined (the lines are vertical), the sides are perpendicular to each other.
To check that the sides of the square are equal in length, we can use the distance formula:
XW = sqrt((6 - 2)^2 + (2 - 2)^2) = sqrt(16) = 4
ZY = sqrt((2 - (-2))^2 + (4 - 4)^2) = sqrt(16) = 4
Since both sides have the same length of 4, all sides of the square are equal in length.
Therefore, the answer is option A) All four sides are equal. The simplest radical form for each side is √40.
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