Question

The mid-point of line segment AB is (2, 4) and the co-ordinates of point A are (5, 7), then the co-ordinates of point B are

• A. (2, -2)
• B. (1, -1)
• C. (-2, -2)
• D. (-1, 1)

Hint:

Think of the midpoint as the average. To find the other end, you can think: "How did I get from A to M?" From x=5 to x=2, you subtract 3. Do it again: 2 - 3 = -1. From y=7 to y=4, you subtract 3. Do it again: 4 - 3 = 1. So, B is (-1, 1).

Answer 1

The Correct Option is D

Step 1: Understanding the Concept:
This problem requires using the midpoint formula to find the coordinates of one endpoint when the other endpoint and the midpoint are known.

Step 2: Key Formula or Approach:
Let the coordinates of A be \((x_1, y_1)\), B be \((x_2, y_2)\), and the midpoint M be \((x_m, y_m)\). The midpoint formula is:
\[ x_m = \frac{x_1 + x_2}{2} \text{and} y_m = \frac{y_1 + y_2}{2} \] We can rearrange this to solve for the unknown endpoint coordinates:
\[ x_2 = 2x_m - x_1 \text{and} y_2 = 2y_m - y_1 \]

Step 3: Detailed Explanation:
We are given:
Midpoint M \((x_m, y_m) = (2, 4)\)
Point A \((x_1, y_1) = (5, 7)\)
We need to find Point B \((x_2, y_2)\).
Using the rearranged formulas:
For the x-coordinate of B:
\[ x_2 = 2(2) - 5 = 4 - 5 = -1 \] For the y-coordinate of B:
\[ y_2 = 2(4) - 7 = 8 - 7 = 1 \] So, the coordinates of point B are \((-1, 1)\).

Step 4: Final Answer:
The co-ordinates of point B are (-1, 1).