Finding a Simple Midpoint

Find the midpoint of the segment with endpoints $A(2, 4)$ and $B(8, 10)$.

Identify the coordinates: $x_1 = 2$, $y_1 = 4$, $x_2 = 8$, $y_2 = 10$ = Coordinates identified

Average the x-coordinates: $\frac{x_1 + x_2}{2} = \frac{2 + 8}{2} = \frac{10}{2} = 5$ = $x_M = 5$

Average the y-coordinates: $\frac{y_1 + y_2}{2} = \frac{4 + 10}{2} = \frac{14}{2} = 7$ = $y_M = 7$

Write the midpoint: $M = (5, 7)$ = Midpoint found

Answer: The midpoint is $(5, 7)$.

Midpoint with Negative Coordinates

Find the midpoint of the segment connecting $P(-4, 3)$ and $Q(6, -1)$.

Identify the coordinates: $x_1 = -4$, $y_1 = 3$, $x_2 = 6$, $y_2 = -1$ = Coordinates identified

Average the x-coordinates: $\frac{-4 + 6}{2} = \frac{2}{2} = 1$ = $x_M = 1$

Average the y-coordinates: $\frac{3 + (-1)}{2} = \frac{2}{2} = 1$ = $y_M = 1$

Write the midpoint: $M = (1, 1)$ = Midpoint found

Answer: The midpoint is $(1, 1)$.

Finding an Endpoint from the Midpoint

Point $A$ is at $(2, 5)$ and the midpoint $M$ is at $(6, 8)$. Find point $B$.

Set up the midpoint equation for x: $\frac{2 + x_2}{2} = 6$ = Equation for $x_2$

Solve for $x_2$: $2 + x_2 = 12 \Rightarrow x_2 = 10$ = $x_2 = 10$

Set up the midpoint equation for y: $\frac{5 + y_2}{2} = 8$ = Equation for $y_2$

Solve for $y_2$: $5 + y_2 = 16 \Rightarrow y_2 = 11$ = $y_2 = 11$

Write the endpoint: $B = (10, 11)$ = Endpoint found

Answer: Point $B$ is at $(10, 11)$.

Mistake: Subtracting coordinates instead of adding them

Why: The midpoint formula requires averaging (adding then dividing by 2), not finding the difference.

Correct: Always ADD the coordinates: $\frac{x_1 + x_2}{2}$, not $\frac{x_2 - x_1}{2}$.

Mistake: Forgetting to divide by 2

Why: Adding the coordinates gives the sum, not the average. You must divide by 2 to find the point in the middle.

Correct: Remember: midpoint means AVERAGE, so always divide the sum by 2.

Mistake: Mixing up x and y coordinates

Why: Adding $x_1$ to $y_2$ (an x-value to a y-value) gives a number that means nothing, and the point comes out in the wrong place.

Correct: Work systematically: first average ALL x-values, then average ALL y-values separately.

Meeting in the Middle

Two friends live in different towns and want to meet exactly halfway between their homes, on the straight line that joins them.

If Alex lives at coordinates $(10, 20)$ and Jordan lives at $(30, 40)$, the meeting point is at $(20, 30)$.

Center of a Playing Field

Sports fields need a center point marked for kickoffs, jump balls, and face-offs.

A rectangular field has corners at $(0, 0)$ and $(100, 60)$. The center is at $(50, 30)$.

Q: Does it matter which point is $(x_1, y_1)$ and which is $(x_2, y_2)$?

A: No! Addition is commutative, so $\frac{x_1 + x_2}{2} = \frac{x_2 + x_1}{2}$. You'll get the same midpoint either way.

Q: What if the midpoint has decimal coordinates?

A: That's perfectly fine! Midpoints often have decimal or fractional coordinates. For example, the midpoint of $(0, 0)$ and $(3, 5)$ is $(1.5, 2.5)$.

Q: How is the midpoint formula related to the average?

A: The midpoint formula IS an average! The x-coordinate of the midpoint is the average of the x-coordinates, and the y-coordinate is the average of the y-coordinates.