Math Snap

STEP 1

1. The coordinates of the points $A$, $B$, and $C$ are given as $A(0,-3)$, $B(4,-2)$, and $C(16,1)$ respectively.
2. Points $A$, $B$, and $C$ are collinear if the area of the triangle formed by these three points is zero.
3. The area of a triangle with vertices $(x_1, y_1)$, $(x_2, y_2)$, $(x_3, y_3)$ can be calculated using the determinant formula.

STEP 2

1. Substitute the coordinates of the points $A$, $B$, and $C$ into the determinant formula for the area of a triangle.
2. Simplify the determinant to determine whether the area is zero.
3. Conclude whether the points are collinear based on the area.

STEP 3

Write the determinant formula for the area of the triangle formed by points $A(0, -3)$, $B(4, -2)$, and $C(16, 1)$.
The area $\Delta$ is given by:
$$\Delta = \frac{1}{2} \left| x_1(y_2 - y_3) + x_2(y_3 - y_1) + x_3(y_1 - y_2) \right|$$ where $(x_1, y_1) = (0, -3)$, $(x_2, y_2) = (4, -2)$, $(x_3, y_3) = (16, 1)$.

STEP 4

Substitute the coordinates of points $A$, $B$, and $C$ into the formula.
$$\Delta = \frac{1}{2} \left| 0(-2 - 1) + 4(1 - (-3)) + 16((-3) - (-2)) \right|$$

STEP 5

Simplify the expression inside the absolute value.
$$\Delta = \frac{1}{2} \left| 0(-3) + 4(1 + 3) + 16(-3 + 2) \right|$$ $$\Delta = \frac{1}{2} \left| 0 + 4 \cdot 4 + 16(-1) \right|$$ $$\Delta = \frac{1}{2} \left| 0 + 16 - 16 \right|$$ $$\Delta = \frac{1}{2} \left| 0 \right|$$ $$\Delta = \frac{1}{2} \cdot 0$$ $$\Delta = 0$$

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