Verify that parallelogram ABCD with vertices A(−5, −1), B(−9, 6), C(−1, 5), and D(3, −2)
is a rhombus. Do the following:
Identify the diagonals of the parallelogram.
Compute the slope of each diagonal.
Explain how you know the parallelogram is a rhombus.
To verify that the parallelogram ABCD is a rhombus, we need to show that it has four congruent sides.
1. Diagonals of the parallelogram:
The diagonals of a parallelogram bisect each other. Let's find the equations of the diagonals using the given vertices:
a) Diagonal AC:
The coordinates of points A and C are A(-5, -1) and C(-1, 5) respectively.
The equation of the diagonal AC can be found using the slope-intercept form:
y - y1 = m(x - x1), where (x1, y1) is a point on the line, and m is the slope.
The slope of AC, mAC = (5 - (-1))/(-1 - (-5)) = 6/4 = 3/2.
Using point A, the equation becomes: y - (-1) = (3/2)(x - (-5)) => y + 1 = (3/2)(x + 5).
b) Diagonal BD:
The coordinates of points B and D are B(-9, 6) and D(3, -2) respectively.
Using the same process, we find the slope of BD, mBD = (-2 - 6)/(3 - (-9)) = -8/12 = -2/3.
Using point B, the equation becomes: y - 6 = (-2/3)(x - (-9)) => y - 6 = (-2/3)(x + 9).
2. Slope of each diagonal:
a) The slope of diagonal AC is 3/2.
b) The slope of diagonal BD is -2/3.
3. Proving the parallelogram is a rhombus:
A rhombus has four congruent sides. To prove this, we need to show that the distance between opposite vertices is the same. We calculate the distances between the given opposite vertices:
a) Distance between A and C:
dAC = sqrt((-1 - (-5))^2 + (5 - (-1))^2) = sqrt(4^2 + 6^2) = sqrt(52) = 2sqrt(13).
b) Distance between B and D:
dBD = sqrt((3 - (-9))^2 + (-2 - 6)^2) = sqrt(12^2 + (-8)^2) = sqrt(144 + 64) = sqrt(208) = 4sqrt(13).
Since dAC = dBD = 2sqrt(13) = 4sqrt(13), we can conclude that all four sides of the parallelogram have the same length. Therefore, the parallelogram ABCD is indeed a rhombus.