The points A (-6,10), B(-4,6) and C(3,-8) are collinear such that
AB `=-(2)/(9)AC`.

Explanation

To determine if the points A (-6, 10), B (-4, 6), and C (3, -8) are collinear and if the relationship \( AB = -\frac{2}{9} AC \) holds true, we can follow these steps:

### Step 1: Check for Collinearity
To check if the points A, B, and C are collinear, we can calculate the area of the triangle formed by these points. If the area is zero, then the points are collinear.

The formula for the area of a triangle given vertices \((x_1, y_1)\), \((x_2, y_2)\), \((x_3, y_3)\) is:
\[ \text{Area} = \frac{1}{2} \left| x_1(y_2 - y_3) + x_2(y_3 - y_1) + x_3(y_1 - y_2) \right| \]

Substituting the coordinates of points A, B, and C:
- \( (x_1, y_1) = (-6, 10) \)
- \( (x_2, y_2) = (-4, 6) \)
- \( (x_3, y_3) = (3, -8) \)

The area becomes:
\[ \text{Area} = \frac{1}{2} \left| -6(6 - (-8)) + (-4)(-8 - 10) + 3(10 - 6) \right| \]

Calculating the terms:
\[ = \frac{1}{2} \left| -6(14) + (-4)(-18) + 3(4) \right| \]
\[ = \frac{1}{2} \left| -84 + 72 + 12 \right| \]
\[ = \frac{1}{2} \left| -84 + 84 \right| = \frac{1}{2} \left| 0 \right| = 0 \]

Since the area is 0, the points A, B, and C are collinear.

### Step 2: Calculate Distances AB and AC
Now we need to calculate the distances \( AB \) and \( AC \).

**Distance AB:**
Using the distance formula:
\[ AB = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \]
Substituting the coordinates of A and B:
\[ AB = \sqrt{((-4) - (-6))^2 + (6 - 10)^2} \]
\[ = \sqrt{(2)^2 + (-4)^2} = \sqrt{4 + 16} = \sqrt{20} = 2\sqrt{5} \]

**Distance AC:**
Using the distance formula:
\[ AC = \sqrt{(x_3 - x_1)^2 + (y_3 - y_1)^2} \]
Substituting the coordinates of A and C:
\[ AC = \sqrt{(3 - (-6))^2 + (-8 - 10)^2} \]
\[ = \sqrt{(9)^2 + (-18)^2} = \sqrt{81 + 324} = \sqrt{405} = 9\sqrt{5} \]

### Step 3: Verify the Relationship
We need to check if \( AB = -\frac{2}{9} AC \):
\[ -\frac{2}{9} AC = -\frac{2}{9} (9\sqrt{5}) = -2\sqrt{5} \]

Now, we compare:
- \( AB = 2\sqrt{5} \)
- \( -\frac{2}{9} AC = -2\sqrt{5} \)

Since \( AB \) is not equal to \( -\frac{2}{9} AC \), the statement is false.

### Conclusion
The points A, B, and C are collinear, but the relationship \( AB = -\frac{2}{9} AC \) is false.

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