The endpoints of one diagonal of a rhombus are (0, -8) and (8, -4). If the coordinates of the 3rd vertex are (1, 0), what are the coordinates of the 4th vertex?

(7, -12)
(7, -8)
(-8, -4)
(-4, -12)

To find the coordinates of the fourth vertex of the rhombus, we need to use the properties of a rhombus.

A rhombus is a special type of quadrilateral in which all sides are equal. The diagonals of a rhombus are perpendicular bisectors of each other, meaning that they intersect at a 90-degree angle, and they divide each other into two equal segments.

Given the coordinates of the endpoints of one diagonal of the rhombus as (0, -8) and (8, -4), we can find the midpoint of this diagonal by averaging the x-coordinates and the y-coordinates separately.

Midpoint X-coordinate = (0 + 8) / 2 = 4
Midpoint Y-coordinate = (-8 + -4) / 2 = -6

Now, we know that the midpoint of the diagonal is (4, -6). Since the diagonals of a rhombus divide each other into two equal segments, we can find the vector that goes from the midpoint of the diagonal to the third vertex (1, 0) by subtracting the coordinates of the midpoint (4, -6) from the coordinates of the third vertex (1, 0).

Vector X-coordinate = 1 - 4 = -3
Vector Y-coordinate = 0 - (-6) = 6

Now, we add this vector to the coordinates of the midpoint to find the fourth vertex.

Fourth vertex X-coordinate = 4 + (-3) = 1
Fourth vertex Y-coordinate = -6 + 6 = 0

Therefore, the coordinates of the fourth vertex are (1, 0).