Question

Show that points A($-1, -1$), B($0, 1$), and C($1, 3$) are collinear.

Hint:

If the slopes of any two pairs of points are equal, the points are collinear.

Answer 1

Step 1: Recall the concept of collinearity.
Three points are collinear if the slopes of any two pairs of points are equal. That is, \[ \text{Slope of } AB = \text{Slope of } BC \] Step 2: Find the slope of AB.
Using the slope formula: \[ m = \frac{y_2 - y_1}{x_2 - x_1} \] For points A($-1, -1$) and B($0, 1$): \[ m_{AB} = \frac{1 - (-1)}{0 - (-1)} = \frac{2}{1} = 2 \] Step 3: Find the slope of BC.
For points B($0, 1$) and C($1, 3$): \[ m_{BC} = \frac{3 - 1}{1 - 0} = \frac{2}{1} = 2 \] Step 4: Compare the slopes.
\[ m_{AB} = m_{BC} = 2 \] Since the slopes are equal, the points A, B, and C are collinear.
Correct Answer: Points A, B, and C are collinear.