The line `(x+6)/5=(y+10)/3=(z+14)/8` is the hypotenuse of an isosceles right-angled triangle whose opposite vertex is `(7,2,4)dot` Then which of the following is not the side of the triangle? a. `(x-7)/2=(y-2)/(-3)=(z-4)/6` b. `(x-7)/3=(y-2)/6=(z-4)/2` c. `(x-7)/3=(y-2)/5=(z-4)/(-1)` d. none of these

Options

A. `(x-7)/(2)=(y-2)/(-3)=(z-4)/(6)`
B. `(x-7)/(3)=(y-2)/(6)=(z-4)/(2)`
C. `(x-7)/(3)=(y-2)/(5)=(z-4)/(-1)`
D. none of these

Explanation

To solve the problem, we need to determine which of the given options does not represent a side of the isosceles right-angled triangle formed with the hypotenuse defined by the line equation.

### Step 1: Understand the Line Equation
The line is given by:
\[ \frac{x + 6}{5} = \frac{y + 10}{3} = \frac{z + 14}{8} \]
Let’s set this equal to a parameter \( t \):
\[ x + 6 = 5t \quad \Rightarrow \quad x = 5t - 6 \]
\[ y + 10 = 3t \quad \Rightarrow \quad y = 3t - 10 \]
\[ z + 14 = 8t \quad \Rightarrow \quad z = 8t - 14 \]

### Step 2: Identify the Opposite Vertex
The opposite vertex of the triangle is given as \( (7, 2, 4) \).

### Step 3: Find the Direction Ratios of the Line
The direction ratios of the line can be extracted from the coefficients of \( t \):
- \( (5, 3, 8) \)

### Step 4: Calculate the Sides of the Triangle
The sides of the triangle must make an angle of \( 45^\circ \) with the hypotenuse. The cosine of \( 45^\circ \) is \( \frac{1}{\sqrt{2}} \).

### Step 5: Check Each Option
Now, we will check each of the given options to see if they can represent the sides of the triangle.

#### Option A:
\[ \frac{x - 7}{2} = \frac{y - 2}{-3} = \frac{z - 4}{6} \]
This represents a line with direction ratios \( (2, -3, 6) \).

#### Option B:
\[ \frac{x - 7}{3} = \frac{y - 2}{6} = \frac{z - 4}{2} \]
This represents a line with direction ratios \( (3, 6, 2) \).

#### Option C:
\[ \frac{x - 7}{3} = \frac{y - 2}{5} = \frac{z - 4}{-1} \]
This represents a line with direction ratios \( (3, 5, -1) \).

### Step 6: Calculate the Dot Products
To determine if these lines make a \( 45^\circ \) angle with the hypotenuse, we will calculate the dot product of their direction ratios with the hypotenuse direction ratios \( (5, 3, 8) \) and check if the cosine of the angle is \( \frac{1}{\sqrt{2}} \).

1. **For Option A:**
\[ (5, 3, 8) \cdot (2, -3, 6) = 5 \cdot 2 + 3 \cdot (-3) + 8 \cdot 6 = 10 - 9 + 48 = 49 \]
The magnitude of \( (2, -3, 6) \):
\[ \sqrt{2^2 + (-3)^2 + 6^2} = \sqrt{4 + 9 + 36} = \sqrt{49} = 7 \]
The magnitude of \( (5, 3, 8) \):
\[ \sqrt{5^2 + 3^2 + 8^2} = \sqrt{25 + 9 + 64} = \sqrt{98} = 7\sqrt{2} \]
Therefore, the cosine:
\[ \cos \theta = \frac{49}{7 \cdot 7\sqrt{2}} = \frac{49}{49\sqrt{2}} = \frac{1}{\sqrt{2}} \]
This is valid.

2. **For Option B:**
\[ (5, 3, 8) \cdot (3, 6, 2) = 5 \cdot 3 + 3 \cdot 6 + 8 \cdot 2 = 15 + 18 + 16 = 49 \]
The magnitude of \( (3, 6, 2) \):
\[ \sqrt{3^2 + 6^2 + 2^2} = \sqrt{9 + 36 + 4} = \sqrt{49} = 7 \]
Therefore, the cosine:
\[ \cos \theta = \frac{49}{7 \cdot 7\sqrt{2}} = \frac{1}{\sqrt{2}} \]
This is valid.

3. **For Option C:**
\[ (5, 3, 8) \cdot (3, 5, -1) = 5 \cdot 3 + 3 \cdot 5 + 8 \cdot (-1) = 15 + 15 - 8 = 22 \]
The magnitude of \( (3, 5, -1) \):
\[ \sqrt{3^2 + 5^2 + (-1)^2} = \sqrt{9 + 25 + 1} = \sqrt{35} \]
Therefore, the cosine:
\[ \cos \theta = \frac{22}{7\sqrt{2} \cdot \sqrt{35}} \neq \frac{1}{\sqrt{2}} \]
This does not satisfy the condition.

### Conclusion
Thus, the option that is not a side of the triangle is:
**Option C: \((x-7)/3=(y-2)/5=(z-4)/(-1)\)**.