Points (-2, 4, 7), (3, -6 -8) and (1, -2, -2) are

Options

A. collinear
B. vertices of an equilateral triangle
C. vertices of an isosceles triangle
D. vertices of a scalene triangle

Answer

To determine whether the points A(-2, 4, 7), B(3, -6, -8), and C(1, -2, -2) are collinear, we will follow these steps:

### Step 1: Identify the Points
Let:
- Point A = (-2, 4, 7)
- Point B = (3, -6, -8)
- Point C = (1, -2, -2)

### Step 2: Calculate the Direction Ratios of AB
To find the direction ratios of the line segment AB, we use the formula:
\[ \text{Direction Ratios of AB} = (x_2 - x_1, y_2 - y_1, z_2 - z_1) \]
Substituting the coordinates of points A and B:
\[ AB = (3 - (-2), -6 - 4, -8 - 7) \]
\[ AB = (3 + 2, -6 - 4, -8 - 7) \]
\[ AB = (5, -10, -15) \]

### Step 3: Calculate the Direction Ratios of BC
Next, we calculate the direction ratios of the line segment BC:
\[ BC = (x_3 - x_2, y_3 - y_2, z_3 - z_2) \]
Substituting the coordinates of points B and C:
\[ BC = (1 - 3, -2 - (-6), -2 - (-8)) \]
\[ BC = (-2, -2 + 6, -2 + 8) \]
\[ BC = (-2, 4, 6) \]

### Step 4: Calculate the Direction Ratios of AC
Now, we calculate the direction ratios of the line segment AC:
\[ AC = (x_3 - x_1, y_3 - y_1, z_3 - z_1) \]
Substituting the coordinates of points A and C:
\[ AC = (1 - (-2), -2 - 4, -2 - 7) \]
\[ AC = (1 + 2, -2 - 4, -2 - 7) \]
\[ AC = (3, -6, -9) \]

### Step 5: Check for Collinearity
To check if the points A, B, and C are collinear, we need to see if the direction ratios of AB, BC, and AC are proportional.

We can express the direction ratios as:
- AB = (5, -10, -15)
- BC = (-2, 4, 6)
- AC = (3, -6, -9)

Now, we can check if:
\[ \frac{AB}{k} = BC \]
or
\[ \frac{AB}{m} = AC \]
for some constants \( k \) and \( m \).

### Step 6: Proportionality Check
1. For AB and BC:
\[ \frac{5}{-2} = \frac{-10}{4} = \frac{-15}{6} \]
Simplifying:
\[ -2.5 = -2.5 = -2.5 \]
This shows that AB and BC are proportional.

2. For AB and AC:
\[ \frac{5}{3} = \frac{-10}{-6} = \frac{-15}{-9} \]
Simplifying:
\[ \frac{5}{3} = \frac{5}{3} = \frac{5}{3} \]
This shows that AB and AC are also proportional.

Since all direction ratios are proportional, we conclude that points A, B, and C are collinear.

### Final Conclusion
Thus, the points A(-2, 4, 7), B(3, -6, -8), and C(1, -2, -2) are collinear.

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