Math Problem Statement

The midpoint of start overline, AB, end overline AB is M, left parenthesis, minus, 1, comma, minus, 5, right parenthesisM(−1,−5). If the coordinates of AA are left parenthesis, 2, comma, minus, 4, right parenthesis(2,−4), what are the coordinates of BB?

Solution

To find the coordinates of point BB, given that M(1,5)M(-1, -5) is the midpoint of AB\overline{AB} and the coordinates of A(2,4)A(2, -4), we can use the midpoint formula. The midpoint formula is given by:

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

Here, M(1,5)M(-1, -5) is the midpoint, A(2,4)A(2, -4) are the coordinates of point AA, and B(x2,y2)B(x_2, y_2) are the coordinates of point BB.

We can set up the following equations based on the midpoint formula:

1=2+x22-1 = \frac{2 + x_2}{2} 5=4+y22-5 = \frac{-4 + y_2}{2}

Now, solve for x2x_2 and y2y_2.

  1. For the x-coordinate: 1=2+x22-1 = \frac{2 + x_2}{2} Multiply both sides by 2: 2=2+x2-2 = 2 + x_2 Subtract 2 from both sides: x2=4x_2 = -4

  2. For the y-coordinate: 5=4+y22-5 = \frac{-4 + y_2}{2} Multiply both sides by 2: 10=4+y2-10 = -4 + y_2 Add 4 to both sides: y2=6y_2 = -6

Therefore, the coordinates of point BB are (4,6)(-4, -6).

Would you like more details or have any questions?

Here are five related questions you might find interesting:

  1. How do you find the distance between two points?
  2. What is the formula for finding the slope of a line segment between two points?
  3. How can you determine if three points are collinear?
  4. What is the equation of the line passing through points A(2,4)A(2, -4) and B(4,6)B(-4, -6)?
  5. How do you find the area of a triangle given the coordinates of its vertices?

Tip: When using the midpoint formula, remember that the midpoint divides the line segment into two equal parts, so it's a great tool to find an unknown endpoint if the other endpoint and the midpoint are known.

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Math Problem Analysis

Mathematical Concepts

Coordinate Geometry
Midpoint Formula

Formulas

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

Theorems

The midpoint theorem

Suitable Grade Level

Grades 8-10