One diagonal of a rhombus has endpoints (-6, 9) and (-2, 1). What are the endpoints of the other diagonal?
To find the endpoints of the other diagonal of a rhombus, we need to use the properties of a rhombus.
A rhombus is a quadrilateral with all four sides of equal length. The diagonals of a rhombus are perpendicular bisectors of each other. This means that the diagonals intersect at a right angle and divide each other into two equal halves.
Let's label the endpoints of the given diagonal as A(-6, 9) and B(-2, 1). We need to find the endpoints of the other diagonal. Let's call these endpoints C and D.
To find the midpoint of the given diagonal, we can use the midpoint formula:
Midpoint = ((x1 + x2) / 2, (y1 + y2) / 2)
Using the given coordinates, the midpoint of AB is:
Midpoint of AB = (((-6) + (-2)) / 2, (9 + 1) / 2)
= (-8 / 2, 10 / 2)
= (-4, 5)
Since the diagonals of a rhombus bisect each other, the midpoint of AC will also be (-4, 5).
Now, let's use the midpoint formula again to find the coordinates of C, one of the endpoints of the other diagonal. We know that the midpoint of AC is (-4, 5) and one endpoint is A(-6, 9). Let's call C(x, y).
Midpoint formula: ((x1 + x2) / 2, (y1 + y2) / 2) = ((-4 + x) / 2, (5 + y) / 2)
Substituting the values into the midpoint formula, we have:
((-6 + x) / 2, (9 + y) / 2) = (-4, 5)
Simplifying this equation, we get:
(-6 + x) / 2 = -4 => -6 + x = -8 => x = -8 + 6 => x = -2
Similarly, (9 + y) / 2 = 5 => 9 + y = 10 => y = 10 - 9 => y = 1
Therefore, C(-2, 1) is one of the endpoints of the other diagonal.
Since the diagonals of a rhombus bisect each other, the other endpoint of the other diagonal will be symmetric to C about the midpoint of the given diagonal.
So, the coordinates of D can be found by reflecting C(-2, 1) about the midpoint (-4, 5).
Using the reflection formula, we get:
D = (2 * (-4) - (-2), 2 * 5 - 1)
= (-8 + 2, 10 - 1)
= (-6, 9)
Thus, the endpoints of the other diagonal are C(-2, 1) and D(-6, 9).
To find the endpoints of the other diagonal of a rhombus, we can use the fact that the diagonals of a rhombus are perpendicular bisectors of each other.
First, let's find the midpoint of the given diagonal. To find the midpoint, we can use the formula:
Midpoint = ((x1 + x2)/2, (y1 + y2)/2)
Substituting the given coordinates:
Midpoint = ((-6 + -2)/2, (9 + 1)/2)
Midpoint = ((-8)/2, (10)/2)
Midpoint = (-4, 5)
Since the diagonals are perpendicular bisectors, the other diagonal will pass through this midpoint. Now, let's find the slope of the given diagonal using the formula:
Slope = (y2 - y1)/(x2 - x1)
Substituting the given coordinates:
Slope = (1 - 9)/(-2 - (-6))
Slope = (-8)/4
Slope = -2
Since the diagonals are perpendicular, the slope of the other diagonal will be the negative reciprocal of -2, which is 1/2.
Now, we have the midpoint (-4, 5) and the slope 1/2 of the other diagonal. We can use point-slope form to find the equation of the other diagonal.
Using the equation:
y - y1 = m(x - x1)
Substituting the midpoint and slope:
y - 5 = (1/2)(x - (-4))
y - 5 = (1/2)(x + 4)
Simplifying the equation:
y - 5 = (1/2)x + 2
y = (1/2)x + 7
Now we have the equation of the other diagonal. To find the endpoints, we can plug in any value for x and solve for y. Let's choose x = 0:
y = (1/2)(0) + 7
y = 7
So, the endpoint of the other diagonal when x = 0 is (0, 7).
Therefore, the endpoints of the other diagonal are (-4, 5) and (0, 7).
the diagonals are perpendicular and bisect each other.
midpoint of the given diagonal is (-4,5), so the other diagonal goes through there.
slope of given diagonal is -8/4 = -2, so the slope of the other diagonal is 1/2.
the sides of a rhombus are all equal, so we need to find (h,k) such that it lies on the line through (-4,5) with slope = 1/2, and the distances from (h,k) to the two given endpoints are the same.
However, that doesn't help much. All points lying on the perpendicular bisector of the given diagonal are equidistant from the diagonal's endpoints!
A rhombus can be anything between a square and a squashed-flat line-like narrow thingy. All we know is that the sides are all the same length.
Without knowledge of the side length, area, or the angle between sides, or something, we are stuck here.