Question

Verify, whether points, A(1, -3), B(2, -5) and C(-4, 7) are collinear or not.

Hint:

Another method to check for collinearity is the distance formula. Calculate the distances AB, BC, and AC. If the sum of two smaller distances equals the largest distance (e.g., AB + BC = AC), the points are collinear. However, the slope method is generally faster.

Answer 1

Step 1: Understanding the Concept:
Three points are collinear if they lie on the same straight line. This can be verified by showing that the slope of the line segment connecting any two points is the same as the slope of the line segment connecting any other two points.

Step 2: Key Formula or Approach:
The slope \(m\) of a line passing through two points \((x_1, y_1)\) and \((x_2, y_2)\) is given by the formula: \[ m = \frac{y_2 - y_1}{x_2 - x_1} \] We will calculate the slope of AB and the slope of BC. If they are equal, the points are collinear.

Step 3: Detailed Explanation:
The given points are A(1, -3), B(2, -5), and C(-4, 7).
1. Find the slope of line segment AB: Here, \((x_1, y_1) = (1, -3)\) and \((x_2, y_2) = (2, -5)\). \[ m_{AB} = \frac{-5 - (-3)}{2 - 1} = \frac{-5 + 3}{1} = \frac{-2}{1} = -2 \] 2. Find the slope of line segment BC: Here, \((x_1, y_1) = (2, -5)\) and \((x_2, y_2) = (-4, 7)\). \[ m_{BC} = \frac{7 - (-5)}{-4 - 2} = \frac{7 + 5}{-6} = \frac{12}{-6} = -2 \] Since the slope of AB is equal to the slope of BC (\(m_{AB} = m_{BC} = -2\)), and they share a common point B, the points A, B, and C lie on the same straight line.

Step 4: Final Answer:
Yes, the points A, B, and C are collinear.