Points (3, 2, 4), (4,5,2), (5,8, 0) are

Options

A. collinear
B. vertices of equilateral triangle
C. vertices of isosceles triangle
D. None of these

Answer

To determine whether the points \( A(3, 2, 4) \), \( B(4, 5, 2) \), and \( C(5, 8, 0) \) are collinear, we will calculate the direction ratios of the lines \( AB \) and \( AC \) and check if they are proportional.

### Step 1: Find the direction ratios of line \( AB \)

The direction ratios of a line formed by two points \( (x_1, y_1, z_1) \) and \( (x_2, y_2, z_2) \) can be calculated using the formula:
\[ \text{Direction Ratios} = (x_2 - x_1, y_2 - y_1, z_2 - z_1) \]

For points \( A(3, 2, 4) \) and \( B(4, 5, 2) \):
\[ \text{Direction Ratios of } AB = (4 - 3, 5 - 2, 2 - 4) = (1, 3, -2) \]

### Step 2: Find the direction ratios of line \( AC \)

Now, we calculate the direction ratios for line \( AC \) using the same formula.

For points \( A(3, 2, 4) \) and \( C(5, 8, 0) \):
\[ \text{Direction Ratios of } AC = (5 - 3, 8 - 2, 0 - 4) = (2, 6, -4) \]

### Step 3: Check if the direction ratios are proportional

To check if the points are collinear, we need to see if the direction ratios of \( AB \) and \( AC \) are proportional. This means we can express them as:
\[ \frac{1}{2} = \frac{3}{6} = \frac{-2}{-4} \]

Calculating each ratio:
- \( \frac{1}{2} = 0.5 \)
- \( \frac{3}{6} = 0.5 \)
- \( \frac{-2}{-4} = 0.5 \)

Since all ratios are equal, the direction ratios are proportional.

### Conclusion

Since the direction ratios of lines \( AB \) and \( AC \) are proportional, the points \( A \), \( B \), and \( C \) are collinear.

### Final Answer

The points \( (3, 2, 4) \), \( (4, 5, 2) \), and \( (5, 8, 0) \) are **collinear**.

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