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 4th vertex of the rhombus, we need to use the properties of a rhombus. One key property of a rhombus is that its diagonals bisect each other at right angles.
Given that the endpoints of one diagonal are (0, -8) and (8, -4), we can find the midpoint of this diagonal. The midpoint can be calculated by taking the average of the x-coordinates and the average of the y-coordinates of the two endpoints.
Midpoint = ((0 + 8) / 2, (-8 + -4) / 2) = (4, -6)
Since the diagonals of a rhombus bisect each other, the midpoint of the diagonal passing through (4, -6) will be the 4th vertex of the rhombus.
Now, let's find the equation of the line passing through the given vertex (1, 0) and the midpoint (4, -6) using the slope-intercept form (y = mx + b), where m is the slope and b is the y-intercept.
Slope (m) = (y2 - y1) / (x2 - x1) = (-6 - 0) / (4 - 1) = -6 / 3 = -2
Using the point-slope form (y - y1 = m(x - x1)), we can substitute the values of x1, y1, and the slope into the equation.
y - 0 = -2(x - 1)
y = -2x + 2
Now, we need to find the x-coordinate of the 4th vertex by substituting the y-coordinate of the midpoint (4, -6) into the equation.
-6 = -2x + 2
-8 = -2x
x = 4
The x-coordinate is 4. Now, we can substitute this value into the equation to find the y-coordinate of the 4th vertex.
y = -2(4) + 2
y = -8 + 2
y = -6
Therefore, the coordinates of the 4th vertex are (4, -6), which corresponds to the option (4, -6).
Simplest way, just count on the grid ...
you have 3 points, and you know that opposite sides must be parallel and equal
A quick calculation shows that each of the given sides is √65 , so far so good.
For the line from (1,0) to (0,-8) we went one to the left, then 8 down
so from (8,-4) go 1 left, then 8 down to get
(7, -11)
(check by taking the lengths of the new lines, I did,
they were both √65)