Math Problem Statement

Find the midpoint, M, of line segment AB where A = (2, 5) and B = (-6, -3).

Solution

The problem requires finding the midpoint, MM, of the line segment AB\overline{AB}, where the coordinates of points AA and BB are given as A(2,5)A(2, 5) and B(6,3)B(-6, -3).

The formula to find the midpoint between two points A(x1,y1)A(x_1, y_1) and B(x2,y2)B(x_2, y_2) is:

M=(x1+x22,y1+y22)M = \left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \right)

Substituting the given coordinates A(2,5)A(2, 5) and B(6,3)B(-6, -3) into the formula:

Mx=2+(6)2=42=2M_x = \frac{2 + (-6)}{2} = \frac{-4}{2} = -2 My=5+(3)2=22=1M_y = \frac{5 + (-3)}{2} = \frac{2}{2} = 1

Thus, the midpoint MM is:

M=(2,1)M = (-2, 1)

Would you like any additional details or have further questions?

Here are five related questions to deepen your understanding:

  1. How would the midpoint formula change in 3D space?
  2. What is the distance between points AA and BB?
  3. How do you interpret the midpoint geometrically?
  4. Can you find the slope of the line segment AB\overline{AB}?
  5. What happens to the midpoint if both points AA and BB move along the same line?

Tip: When calculating midpoints, always ensure you handle both positive and negative signs carefully to avoid errors.

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Math Problem Analysis

Mathematical Concepts

Coordinate Geometry
Midpoint Calculation

Formulas

Midpoint Formula: M = ((x1 + x2) / 2, (y1 + y2) / 2)

Theorems

Midpoint Theorem

Suitable Grade Level

Grades 6-8