Show that the quadrilateral with verticies at P(0,2,5), Q(1,6,2), R(7,4,2), and S(6,0,5) is a parallelogram.
vector PQ = 1 i + 4 j - 3 k
vector SR = 1 i + 4 j - 3 k
parallel
vector QR = 6 i - 2 j + 0 k
Vector PS = 6 i - 2 j + 0 k
parallel
done
Your figure is embedded in three dimensional space, which makes it harder.
First consider the side lengths:
PQ: sqrt(1^2 + 4^2 + 3^2) = 5
QR: sqrt(6^2 + 2^2 + 0) = sqrt40
RS: sqrt(1^2 + 4^2 + 3^2) = 5
SP: sqrt(6^2 + 2^2 + 0) = sqrt40
Next look at the direction cosines. They are the same for the PQ and RS pair, and for the pair QR and SP.
Opposite sides are of equal length and parallel. Adjacent sides are connected. It must be a parallelogram. There must be a theorem for that.
To show that the quadrilateral with vertices at P(0,2,5), Q(1,6,2), R(7,4,2), and S(6,0,5) is a parallelogram, we need to demonstrate that its opposite sides are parallel.
To determine if two lines are parallel, we need to check if their directional vectors are proportional. The directional vector of a line is obtained by subtracting the coordinates of the two points on the line.
Let's start by finding the directional vectors of the two opposite sides of the quadrilateral:
Side PQ:
Directional vector PQ = Q - P = (1 - 0, 6 - 2, 2 - 5) = (1, 4, -3)
Side RS:
Directional vector RS = S - R = (6 - 7, 0 - 4, 5 - 2) = (-1, -4, 3)
Now, let's compare the components of the directional vectors. If the ratios of the corresponding components are equal, then the sides are parallel.
Comparing the x-component: 1 / -1 = -1
Comparing the y-component: 4 / -4 = -1
Comparing the z-component: -3 / 3 = -1
Since the ratios of the corresponding components are all equal (-1), we can conclude that opposite sides PQ and RS are parallel.
Now, let's check the other pair of opposite sides:
Side QR:
Directional vector QR = R - Q = (7 - 1, 4 - 6, 2 - 2) = (6, -2, 0)
Side PS:
Directional vector PS = S - P = (6 - 0, 0 - 2, 5 - 5) = (6, -2, 0)
Comparing the x-component: 6 / 6 = 1
Comparing the y-component: -2 / -2 = 1
Comparing the z-component: 0 / 0 = 0
Again, the ratios of the corresponding components are all equal (1), so we can conclude that the other pair of opposite sides QR and PS are parallel.
Since both pairs of opposite sides are parallel, we have shown that the quadrilateral with vertices at P(0,2,5), Q(1,6,2), R(7,4,2), and S(6,0,5) is a parallelogram.