Question

The points A(4, -2, 1), B(7, -4, 7), C(2, -5, 10), and D(-1, -3, 4) are the vertices of a:

• A. Tetrahedron
• B. Parallelogram
• C. Rhombus
• D. Square

Hint:

To determine the type of quadrilateral formed by four points in 3D space, calculate the vectors for the sides and check if opposite sides are equal and parallel. This indicates a parallelogram.

Answer 1

The Correct Option is B

Step 1: Understanding the Problem
We are given four points in 3D space: A(4, -2, 1), B(7, -4, 7), C(2, -5, 10), and D(-1, -3, 4). We need to determine the type of figure formed by these points.
Step 2: Calculating the Vectors
Calculate the vectors for the sides: \[ \vec{AB} = B - A = (7-4, -4-(-2), 7-1) = (3, -2, 6), \] \[ \vec{AD} = D - A = (-1-4, -3-(-2), 4-1) = (-5, -1, 3), \] \[ \vec{BC} = C - B = (2-7, -5-(-4), 10-7) = (-5, -1, 3), \] \[ \vec{DC} = C - D = (2-(-1), -5-(-3), 10-4) = (3, -2, 6). \]
Step 3: Checking for Parallelogram
A quadrilateral is a parallelogram if opposite sides are equal and parallel.
Here, \( \vec{AB} = \vec{DC} \) and \( \vec{AD} = \vec{BC} \), which indicates that opposite sides are equal and parallel.
Step 4: Verifying Other Options
Tetrahedron: A tetrahedron has four triangular faces, which is not the case here.
Rhombus: A rhombus has all sides equal, which is not verified here.
Square: A square has all sides equal and all angles 90 degrees, which is not verified here.
Step 5: Matching with the Options
The figure formed by the points is a parallelogram, which corresponds to option (B). Final Answer: The points form a (B) Parallelogram.