Midpoint of points

The midpoint of two points is the point that is exactly halfway between them.

Importantly, this means that the x and y coordinates of the midpoint are the averages of the x and y coordinates of the two points respectively.

Finding the midpoint of two points

To find the midpoint of the two points, we can add up the x-coordinates of both points, divide by 2 to get the average x-coordinate, and do the same for the y-coordinates.

Or, use the formula:

\text{Midpoint} = ( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} )

We can also separate the formula into its x and y components (x_M and y_M are the x and y coordinates of the midpoint respectively):

x_M = \frac{x_1 + x_2}{2}

y_M = \frac{y_1 + y_2}{2}

Find the midpoint of the points (2, 3) and (6, 7).

• Values we know:
  • x_1 = 2
  • y_1 = 3
  • x_2 = 6
  • y_2 = 7
• Substitute into the formula:
  • \text{Midpoint} = ( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} )
  • = ( \frac{2 + 6}{2}, \frac{3 + 7}{2} )
  • = ( \frac{8}{2}, \frac{10}{2} )
  • = (4, 5)
• Answer: (4, 5).

Find the midpoint of the points (-1, 4) and (3, -2).

• Values we know:
  • x_1 = -1
  • y_1 = 4
  • x_2 = 3
  • y_2 = -2
• Substitute into the formula:
  • \text{Midpoint} = ( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} )
  • = ( \frac{-1 + 3}{2}, \frac{4 + (-2)}{2} )
  • = ( \frac{2}{2}, \frac{2}{2} )
  • = (1, 1)
• Answer: (1, 1).

Find the midpoint of the points (0, 0) and (5, 10).

• Values we know:
  • x_1 = 0
  • y_1 = 0
  • x_2 = 5
  • y_2 = 10
• Substitute into the formula:
  • \text{Midpoint} = ( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} )
  • = ( \frac{0 + 5}{2}, \frac{0 + 10}{2} )
  • = ( \frac{5}{2}, \frac{10}{2} )
  • = (2.5, 5)
• Answer: (2.5, 5).

Find the midpoint of the points (-4, -6) and (4, 6).

• Values we know:
  • x_1 = -4
  • y_1 = -6
  • x_2 = 4
  • y_2 = 6
• Substitute into the formula:
  • \text{Midpoint} = ( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} )
  • = ( \frac{-4 + 4}{2}, \frac{-6 + 6}{2} )
  • = ( \frac{0}{2}, \frac{0}{2} )
  • = (0, 0)
• Answer: (0, 0).

Finding one point given the other point and the midpoint

Find the point A, given that the midpoint between point A and point B (4, 8) is (6, 10).

• Values we know:
  • Midpoint = (6, 10)
  • x_2 = 4
  • y_2 = 8
• Using the midpoint formula, we can set up two equations:
  • 6 = \frac{x_1 + 4}{2}
  • 10 = \frac{y_1 + 8}{2}
• Solving for x_1:
  • 6 \times 2 = x_1 + 4
  • 12 = x_1 + 4
  • x_1 = 12 - 4
  • x_1 = 8
• Solving for y_1:
  • 10 \times 2 = y_1 + 8
  • 20 = y_1 + 8
  • y_1 = 20 - 8
  • y_1 = 12
• Answer: Point A is (8, 12).

Find the point C, given that the midpoint between point C and point D (2, -4) is (5, 1).

• Values we know:
  • Midpoint = (5, 1)
  • x_2 = 2
  • y_2 = -4
• Using the midpoint formula, we can set up two equations:
  • 5 = \frac{x_1 + 2}{2}
  • 1 = \frac{y_1 + (-4)}{2}
• Solving for x_1:
  • 5 \times 2 = x_1 + 2
  • 10 = x_1 + 2
  • x_1 = 10 - 2
  • x_1 = 8
• Solving for y_1:
  • 1 \times 2 = y_1 - 4
  • 2 = y_1 - 4
  • y_1 = 2 + 4
  • y_1 = 6
• Answer: Point C is (8, 6).