State whether the following statements are true or false. Justify your answer:
The points A (-6,10), B(-4,6) and C(3,-8) are collinear such that
`AB` `=(2)/(9)AC`.

Explanation

To determine whether the points A (-6, 10), B (-4, 6), and C (3, -8) are collinear and whether the statement \( AB = \frac{2}{9} AC \) is true, we will 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 triangle ABC using the formula:

\[ \text{Area} = \frac{1}{2} \left| x_1(y_2 - y_3) + x_2(y_3 - y_1) + x_3(y_1 - y_2) \right| \]

Where:
- \( A(x_1, y_1) = (-6, 10) \)
- \( B(x_2, y_2) = (-4, 6) \)
- \( C(x_3, y_3) = (3, -8) \)

Substituting the coordinates into the formula:

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

Calculating each term:

1. \( -6(6 + 8) = -6 \times 14 = -84 \)
2. \( -4(-8 - 10) = -4 \times -18 = 72 \)
3. \( 3(10 - 6) = 3 \times 4 = 12 \)

Now, substituting back into the area formula:

\[ \text{Area} = \frac{1}{2} \left| -84 + 72 + 12 \right| = \frac{1}{2} \left| -84 + 84 \right| = \frac{1}{2} \left| 0 \right| = 0 \]

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

### Step 2: Calculate Distances AB and AC
Next, we need to calculate the lengths of segments 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 again:

\[ AC = \sqrt{(x_3 - x_1)^2 + (y_3 - y_1)^2} \]

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 \( AB = \frac{2}{9} AC \)

Now we need to check if \( AB = \frac{2}{9} AC \):

Calculating the right-hand side:

\[ \frac{2}{9} AC = \frac{2}{9} \times 9\sqrt{5} = 2\sqrt{5} \]

Since \( AB = 2\sqrt{5} \), we find that:

\[ AB = \frac{2}{9} AC \]

### Conclusion
Both statements are true. The points A, B, and C are collinear, and \( AB = \frac{2}{9} AC \).