Math Problem Statement

The midpoint of start overline, AB, end overline AB is M, left parenthesis, minus, 6, comma, 3, right parenthesisM(−6,3). If the coordinates of AA are left parenthesis, minus, 5, comma, 5, right parenthesis(−5,5), what are the coordinates of BB?

Solution

To find the coordinates of point B(x2,y2)B(x_2, y_2), we can use the midpoint formula. The midpoint M(xm,ym)M(x_m, y_m) of a line segment between points A(x1,y1)A(x_1, y_1) and B(x2,y2)B(x_2, y_2) is given by:

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

We know:

  • M=(6,3)M = (-6, 3)
  • A=(5,5)A = (-5, 5)

Using the midpoint formula for each coordinate:

For the x-coordinate:

6=5+x22-6 = \frac{-5 + x_2}{2}

Multiply both sides by 2: 12=5+x2-12 = -5 + x_2

Solve for x2x_2: x2=12+5=7x_2 = -12 + 5 = -7

For the y-coordinate:

3=5+y223 = \frac{5 + y_2}{2}

Multiply both sides by 2: 6=5+y26 = 5 + y_2

Solve for y2y_2: y2=65=1y_2 = 6 - 5 = 1

Final Answer:

The coordinates of BB are (7,1)(-7, 1).

Would you like more details on this process?


Here are 5 related questions to explore:

  1. How do you find the midpoint of a line segment when both endpoints are given?
  2. What is the geometric significance of the midpoint in a coordinate plane?
  3. How can the midpoint formula be used to check if a line segment is bisected?
  4. If you change the coordinates of point AA, how does it affect the midpoint?
  5. How would you find the midpoint if one of the points is a variable, like A(x1,y1)A(x_1, y_1)?

Tip: Always double-check your work by plugging your calculated point back into the midpoint formula to verify accuracy.

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Math Problem Analysis

Mathematical Concepts

Coordinate Geometry
Midpoint Formula

Formulas

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

Theorems

Midpoint Theorem

Suitable Grade Level

Grades 6-8