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: Accidentally pairing $x_1$ with $y_2$ or making calculation errors.

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 at a point equidistant from both.

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.