Question
Download Solution PDFConsider the points A(2, 4, 6), B(−2, −4, −2), C(4, 6, 4), and D(8, 14, 12). Which of the following statements is/are correct?
1. The points are the vertices of a rectangle ABCD.
2. The mid-point of A C is the same as that of BD.
Select the correct answer using the code given below :
Answer (Detailed Solution Below)
Detailed Solution
Download Solution PDFCalculation:
Statement 1: The points are the vertices of a rectangle ABCD.
The points A(2, 4, 6), B(−2, −4, −2), C(4, 6, 4), and D(8, 14, 12).
AB = 12
CD = 12
Hence, AB = CD and BC = DA
So, it could be a parallelogram or a rectangle. To confirm this, we will find the dot product of the vector
Refering to the diagram,
Now,
So, AB & CD are not perpendicular
So, ABCD is a parallelogram.
Therefore, The points are not the vertices of a rectangle ABCD. It is forming the parallelogram.
Hence, statement 1 is incorrect.
Statement 2: The mid-point of A C is the same as that of BD.
The mid-point of AC =
= (3, 5, 5)
The mid-point of BD =
= (3, 5, 5)
Hence, The mid-point of A C is the same as that of BD.
So, the statement 2 is correct.
∴ The correct option is (2)
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