Math Problem Statement

The midpoint of start overline, AB, end overline AB is M, left parenthesis, minus, 4, comma, 0, right parenthesisM(−4,0). If the coordinates of AA are left parenthesis, minus, 7, comma, 3, right parenthesis(−7,3), what are the coordinates of BB?

Solution

To find the coordinates of point BB, we use the formula for the midpoint between two points. The midpoint M(xm,ym)M(x_m, y_m) of a line segment connecting points A(xa,ya)A(x_a, y_a) and B(xb,yb)B(x_b, y_b) is given by:

M(xm,ym)=(xa+xb2,ya+yb2)M(x_m, y_m) = \left( \frac{x_a + x_b}{2}, \frac{y_a + y_b}{2} \right)

We know:

  • The coordinates of MM are (4,0)(-4, 0),
  • The coordinates of AA are (7,3)(-7, 3).

Let B(xb,yb)B(x_b, y_b) represent the unknown coordinates of point BB.

Using the midpoint formula:

4=7+xb2and0=3+yb2-4 = \frac{-7 + x_b}{2} \quad \text{and} \quad 0 = \frac{3 + y_b}{2}

Now, solve each equation:

For the xx-coordinate:

4=7+xb2-4 = \frac{-7 + x_b}{2} Multiply both sides by 2: 8=7+xb-8 = -7 + x_b Add 7 to both sides: xb=1x_b = -1

For the yy-coordinate:

0=3+yb20 = \frac{3 + y_b}{2} Multiply both sides by 2: 0=3+yb0 = 3 + y_b Subtract 3 from both sides: yb=3y_b = -3

Thus, the coordinates of BB are (1,3)(-1, -3).

Would you like more details or have any questions? Here are 5 related questions to deepen your understanding:

  1. How do you derive the midpoint formula?
  2. What happens if one or both coordinates of MM are negative?
  3. Can you apply this method to 3-dimensional points?
  4. How can you verify your answer using the distance formula?
  5. What if the midpoint is given in terms of variables?

Tip: Always check your work by substituting back into the original equations to ensure consistency.

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Math Problem Analysis

Mathematical Concepts

Geometry
Coordinate Geometry

Formulas

Midpoint formula: M(x_m, y_m) = ((x_a + x_b)/2, (y_a + y_b)/2)

Theorems

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Suitable Grade Level

Grades 8-10