7. ABC has vertices A(3,2,-5), B(4,-1,7) and C(-8,3,-6).


a) Determine the area of ABC.


b) Determine the coordinates of point D such that ABCD is a parallelogram.

c) Is the parallelogram a rectangle? Justify your answer.

for (a), determine the side lengths. If it's a right triangle, the area is easy. If not, you can use Heron's formula, or evaluate a.bxc where a,b,c are the vectors for the sides.

for (b) there are several ways to pick D. If ABC is a right triangle, one of those ways will produce a rectangle. Otherwise, not.

To solve these questions, we can use the properties and formulas related to triangles and parallelograms.

a) To determine the area of triangle ABC, we can use the formula for the area of a triangle formed by three 3D points:

Area = 1/2 * |(x1(y2 - y3) + x2(y3 - y1) + x3(y1 - y2))|

Given the coordinates of points A, B, C as A(3,2,-5), B(4,-1,7), and C(-8,3,-6), we can substitute these values into the formula and calculate the area.

Area = 1/2 * |(3((-1) - 3) + 4(3 - 2) + (-8)(2 - (-1)))|
= 1/2 * |(-3 + 4 + 24)|
= 1/2 * 25
= 12.5 square units

Therefore, the area of triangle ABC is 12.5 square units.

b) To determine the coordinates of point D such that ABCD is a parallelogram, we can use the fact that opposite sides of a parallelogram are equal in length and parallel to each other.

We already have three points A, B, and C. To find point D, we can simply take the opposite side of point B with respect to point A, and add it to point C.

Direction vector of AB = (4 - 3, -1 - 2, 7 - (-5)) = (1, -3, 12)
Coordinates of D = C + AB = (-8, 3, -6) + (1, -3, 12) = (-7, 0, 6)

Therefore, the coordinates of point D are (-7, 0, 6) such that ABCD is a parallelogram.

c) To determine if the parallelogram ABCD is a rectangle, we need to check if the diagonals are perpendicular.

We can find the vectors representing the diagonals BD and AC:

Vector BD = D - B = (-7, 0, 6) - (4, -1, 7) = (-11, 1, -1)
Vector AC = C - A = (-8, 3, -6) - (3, 2, -5) = (-11, 1, -1)

Since the vectors BD and AC are identical, the diagonals of the parallelogram are the same.

To check if the diagonals are perpendicular, we can calculate the dot product between the vectors BD and AC. If the dot product is zero, the diagonals are perpendicular.

Dot product of BD and AC = (-11)(-11) + 1 * 1 + (-1)(-1) = 121 + 1 + 1 = 123

Since the dot product is not zero, the diagonals are not perpendicular, and therefore the parallelogram ABCD is not a rectangle.

Therefore, the parallelogram ABCD is not a rectangle.

a) To determine the area of triangle ABC, we can use the formula for the magnitude of the cross product of two vectors.

First, we need to find two vectors that are in the plane of triangle ABC. Let's choose vectors AB and AC.

Vector AB can be found by subtracting the coordinates of point A from the coordinates of point B:
AB = B - A = (4, -1, 7) - (3, 2, -5) = (1, -3, 12)

Vector AC can be found by subtracting the coordinates of point A from the coordinates of point C:
AC = C - A = (-8, 3, -6) - (3, 2, -5) = (-11, 1, -1)

Next, we can calculate the cross product of vectors AB and AC:
AB x AC = (1, -3, 12) x (-11, 1, -1) = [(12)(-1) - (-3)(-1), (1)(-1) - (12)(-11), (1)(-3) - (1)(-11)]
= (-9, -133, 8)

The magnitude of this cross product gives us the area of triangle ABC:
Area = ||AB x AC|| = √((-9)^2 + (-133)^2 + 8^2) = √(81 + 17689 + 64) = √17834 ≈ 133.60

Therefore, the area of triangle ABC is approximately 133.60 square units.

b) To determine the coordinates of point D such that ABCD is a parallelogram, we need to find a vector that is parallel to AB and has the same magnitude as AC.

One way to accomplish this is to add vector AC to the coordinates of point B:
D = B + AC = (4, -1, 7) + (-11, 1, -1) = (-7, 0, 6)

Therefore, the coordinates of point D are (-7, 0, 6) such that ABCD is a parallelogram.

c) To determine if the parallelogram ABCD is a rectangle, we need to check if the diagonals are perpendicular to each other.

The diagonals of ABCD are AD and BC. Let's find the vectors for these diagonals:

Vector AD = D - A = (-7, 0, 6) - (3, 2, -5) = (-10, -2, 11)
Vector BC = C - B = (-8, 3, -6) - (4, -1, 7) = (-12, 4, -13)

Now, we can calculate the dot product of vectors AD and BC. If the dot product is 0, then the diagonals are perpendicular.

AD · BC = (-10)(-12) + (-2)(4) + (11)(-13) = 120 - 8 - 143 = -31

Since the dot product is not 0, the diagonals AD and BC are not perpendicular. Therefore, the parallelogram ABCD is not a rectangle.

And that's how we determine the area of triangle ABC, find the coordinates of point D to form a parallelogram, and check if the parallelogram is a rectangle.