Let `M=[(a,b,c),(d,e,f),(1,1,1)] and N=(M^(2))/(2)`. If `(a-b)^(2)+(d-e)^(2)=36,`
`(b-c)^(2)+(e-f)^(2)=64,`
`(a-c)^(2)+(d-f)^(2)=100`, then value of `|N|` is equal to

Options

A. 1152
B. 48
C. 144
D. 288

Answer

To solve the problem, we need to find the value of \(|N|\) given the matrix \(M\) and the conditions provided. Let's break it down step by step.

### Step 1: Understand the Matrix and Given Conditions
We have the matrix:
\[ M = \begin{pmatrix} a & b & c \\ d & e & f \\ 1 & 1 & 1 \end{pmatrix} \]
We are given three equations:
1. \((a-b)^2 + (d-e)^2 = 36\)
2. \((b-c)^2 + (e-f)^2 = 64\)
3. \((a-c)^2 + (d-f)^2 = 100\)

### Step 2: Interpret the Equations
These equations can be interpreted as distances between points in a coordinate system:
- The first equation represents the distance between points \((a, d)\) and \((b, e)\).
- The second equation represents the distance between points \((b, e)\) and \((c, f)\).
- The third equation represents the distance between points \((a, d)\) and \((c, f)\).

### Step 3: Calculate the Distances
From the equations, we can find the lengths of the sides of the triangle formed by these points:
- Let \(AB = \sqrt{36} = 6\)
- Let \(BC = \sqrt{64} = 8\)
- Let \(AC = \sqrt{100} = 10\)

### Step 4: Verify Right Triangle Condition
We check if these sides satisfy the Pythagorean theorem:
\[ AB^2 + BC^2 = 6^2 + 8^2 = 36 + 64 = 100 = AC^2 \]
Since this holds true, triangle \(ABC\) is a right triangle.

### Step 5: Calculate the Area of Triangle ABC
The area \(A\) of a right triangle can be calculated using:
\[ A = \frac{1}{2} \times \text{base} \times \text{height} \]
Taking \(AB\) as the base and \(BC\) as the height:
\[ A = \frac{1}{2} \times 6 \times 8 = 24 \]

### Step 6: Relate Area to Determinant of Matrix M
The area of triangle \(ABC\) can also be expressed in terms of the determinant of matrix \(M\):
\[ \text{Area} = \frac{1}{2} | \text{det}(M) | \]
Thus, we have:
\[ 24 = \frac{1}{2} | \text{det}(M) | \]
This implies:
\[ | \text{det}(M) | = 48 \]

### Step 7: Calculate the Determinant of N
We know that:
\[ N = \frac{M^2}{2} \]
Using the property of determinants:
\[ |N| = \left| \frac{M^2}{2} \right| = \frac{1}{2^3} |M^2| = \frac{1}{8} |M|^2 \]
Since \(|M| = 48\), we find:
\[ |M^2| = |M|^2 = 48^2 = 2304 \]
Thus:
\[ |N| = \frac{1}{8} \times 2304 = 288 \]

### Final Answer
The value of \(|N|\) is:
\[ \boxed{288} \]