The endpoints of one diagonal of a The 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)

sketch graph

from y = -8 must go down 4
so at y = -12
from x = 0 must go right 7
so at x = 7
so
(7,-12)

To find the coordinates of the fourth vertex of the rhombus, we need to understand the properties of a rhombus. A rhombus is a quadrilateral with all sides equal in length and opposite angles equal in measure.

Given that the coordinates of the third vertex are (1, 0), we can see that this point lies on one of the diagonals of the rhombus. So, to find the coordinates of the fourth vertex, we first need to determine the length and direction of the diagonals.

Using the given coordinates of the endpoints of one diagonal, we can find the length and direction of the diagonal by subtracting the coordinates. Let's calculate:

Length of the diagonal = sqrt((8 - 0)^2 + (-4 - (-8))^2)
= sqrt(8^2 + 4^2)
= sqrt(64 + 16)
= sqrt(80)
= 4sqrt(5)

Next, we need to find the slope of this diagonal. The slope can be found by subtracting the y-coordinates and dividing it by the difference in the x-coordinates:

Slope of the diagonal = (-4 - (-8)) / (8 - 0)
= (-4 + 8) / 8
= 4/8
= 1/2

The slope of one diagonal is 1/2. Since a rhombus has diagonals that are perpendicular to each other, the slope of the other diagonal is the negative reciprocal of 1/2, which is -2.

Now that we know the slope of the other diagonal, we can find its equation using the point-slope form:

y - 0 = -2(x - 1)
y = -2x + 2

To find the coordinates of the fourth vertex, we need to find the intersection point of this diagonal equation with the given diagonal. Setting the equations equal to each other, we get:

-2x + 2 = 2x - 8
4x = 10
x = 10/4
x = 5/2

Substituting this value for x back into either of the equations, we can find the corresponding y-coordinate:

y = -2(5/2) + 2
y = -5 + 2
y = -3

Therefore, the coordinates of the fourth vertex are (5/2, -3).

To convert this fraction into decimal form, we can divide the numerator by the denominator: 5 ÷ 2 = 2.5. So, the coordinates of the fourth vertex are approximately (2.5, -3).

Therefore, the correct option is not listed among the answer choices.