To prove that points $A(2,3)$, $B(1,4)$, and $C(3,8)$ are collinear, we will compute the slopes of the line segments $AB$ and $AC$. If the slopes are equal, then the three points lie on the same straight line.

Step 1: Formula for the slope

The slope of a line passing through two points $(x_1, y_1)$ and $(x_2, y_2)$ is given by: $m = \frac{y_2 - y_1}{x_2 - x_1}$

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Step 2: Slope of $AB$

The coordinates of $A$ and $B$ are $A(2,3)$ and $B(1,4)$. $m_{AB} = \frac{4 - 3}{1 - 2} = \frac{1}{-1} = -1$

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Step 3: Slope of $AC$

The coordinates of $A$ and $C$ are $A(2,3)$ and $C(3,8)$. $m_{AC} = \frac{8 - 3}{3 - 2} = \frac{5}{1} = 5$

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Step 4: Compare $m_{AB}$ and $m_{AC}$

We observe that: $m_{AB} = -1 \quad \text{and} \quad m_{AC} = 5$

Since the slopes are not equal, the points $A$, $B$, and $C$ are not collinear.

Would you like to review this step further or solve a similar problem? Here are some questions for exploration:

1. What happens if we calculate the slope of $BC$?
2. How can we verify collinearity using the area of a triangle formed by three points?
3. Can you find a line equation passing through $A$ and $B$? Does $C$ satisfy the equation?
4. How do slopes help identify parallel and perpendicular lines?
5. How can we confirm if two lines intersect using their slopes?

Tip: If you struggle to verify collinearity, remember that three points are collinear if the area of the triangle they form is zero.