6. Midpoint and Special Points

The midpoint between two points is the point exactly halfway between them. Use the midpoint formula:

\[ M = \left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \right) \]

Simply average the x-coordinates and average the y-coordinates.

Worked Example 1: Find the midpoint between \((2, 4)\) and \((8, 10)\).

\[ M = \left( \frac{2 + 8}{2}, \frac{4 + 10}{2} \right) = \left( \frac{10}{2}, \frac{14}{2} \right) = (5, 7) \]

Check: The distance from \((2, 4)\) to \((5, 7)\) is \(\sqrt{3^2 + 3^2} = \sqrt{18} \approx 4.24\), and the distance from \((5, 7)\) to \((8, 10)\) is also \(\sqrt{3^2 + 3^2} \approx 4.24\). ✓

Worked Example 2: Find the midpoint between \((-3, 5)\) and \((1, -3)\).

\[ M = \left( \frac{-3 + 1}{2}, \frac{5 + (-3)}{2} \right) = \left( \frac{-2}{2}, \frac{2}{2} \right) = (-1, 1) \]

Real-world context: Two cities are located at \((10, 20)\) and \((50, 80)\) on a coordinate map. A supply depot should be built at the midpoint:
\[ M = \left( \frac{10 + 50}{2}, \frac{20 + 80}{2} \right) = (30, 50) \]

Common mistake: forgetting to divide by 2. The midpoint is not \((x_1 + x_2, y_1 + y_2)\).

💡 Tip: The midpoint is useful for finding centers of circles, balanced points in designs, and equal-distance locations.

Midpoint as an Average Position

The midpoint of a segment is the point exactly halfway between its endpoints. If \(A(x_1,y_1)\) and \(B(x_2,y_2)\), halfway in the horizontal direction is the average of the x-coordinates, and halfway in the vertical direction is the average of the y-coordinates:
\[ M=\left(\frac{x_1+x_2}{2},\frac{y_1+y_2}{2}\right). \]
This formula is two number-line midpoint calculations performed at the same time. Averaging is appropriate because the midpoint must balance the two endpoints in each coordinate direction.

Worked example: Find the midpoint of \(A(-7,5)\) and \(B(3,-9)\).
\[ M=\left(\frac{-7+3}{2},\frac{5+(-9)}{2}\right) =\left(\frac{-4}{2},\frac{-4}{2}\right)=(-2,-2). \]
Check through changes: from A to M, the change is \((+5,-7)\); from M to B, it is also \((+5,-7)\). Equal coordinate changes prove M bisects the segment.

Why the Formula Works

Let \(M=(h,k)\). If M is halfway between the x-values, then its directed change from \(x_1\) equals the change from h to \(x_2\):
\[ h-x_1=x_2-h. \]
Solving gives \(2h=x_1+x_2\), hence \(h=(x_1+x_2)/2\). The same argument gives \(k=(y_1+y_2)/2\). This derivation shows that midpoint is not “add everything and divide by two”; x-values are averaged together and y-values are averaged together.

The midpoint can have fractions or decimals even when both endpoints have integer coordinates. Between \((0,0)\) and \((5,3)\), the midpoint is \((2.5,1.5)\). A midpoint need not be a visible grid intersection.

Finding a Missing Endpoint

If one endpoint and the midpoint are known, reverse the averaging process. From
\[ m_x=\frac{x_1+x_2}{2}, \]
multiply by 2 and subtract the known coordinate:
\[ x_2=2m_x-x_1. \]
Similarly, \(y_2=2m_y-y_1\).

Worked example: Endpoint A is \((-4,7)\), and midpoint M is \((3,1)\). Find endpoint B.
\[ x_B=2(3)-(-4)=10, \qquad y_B=2(1)-7=-5. \]
Thus \(B=(10,-5)\). Verify by averaging \((-4,7)\) and \((10,-5)\): the result is \((3,1)\).

A visual method gives the same result. From A to M, move 7 right and 6 down. Repeat that same directed movement from M to B: \((3+7,1-6)=(10,-5)\).

Midpoint, Distance, and Slope

The midpoint lies on the original segment and divides it into two equal lengths. For A, M, and B above, the vector from A to M is half the vector from A to B. Therefore
\[ AM=MB=\frac12AB. \]
The three points also share the same slope whenever the segment is nonvertical. These properties provide independent checks.

For \(A(1,2)\), \(B(9,8)\), and \(M(5,5)\),
\[ AB=\sqrt{8^2+6^2}=10, \]
while
\[ AM=\sqrt{4^2+3^2}=5. \]
The slope from A to B and from A to M is \(6/8=3/4\). Midpoint, distance, and slope agree.

Segment Division and Weighted Averages

The midpoint is a 1:1 division point. Sometimes a context asks for a point one-quarter or three-quarters of the way from A to B. Use the total coordinate change and take the required fraction.

From \(A(-2,4)\) to \(B(10,-8)\), the total change is \((12,-12)\). One-quarter of the way from A is
\[ (-2,4)+\frac14(12,-12)=(-2,4)+(3,-3)=(1,1). \]
Three-quarters of the way is \((-2,4)+(9,-9)=(7,-5)\). The midpoint corresponds to one-half: \((-2,4)+(6,-6)=(4,-2)\).

This vector approach is useful in map and design problems because it retains direction. It also explains interpolation: estimating a location or value between two known points.

Geometric Applications

Diagonals of a parallelogram bisect each other. Therefore, if the diagonals of a quadrilateral have the same midpoint, that is evidence the quadrilateral is a parallelogram. For vertices \(A(0,0)\), \(B(6,2)\), \(C(8,8)\), and \(D(2,6)\), midpoint of AC is \((4,4)\), and midpoint of BD is also \((4,4)\). The diagonals bisect each other.

The center of a circle is also the midpoint of any diameter. If endpoints of a diameter are \((-5,1)\) and \((7,9)\), the center is \((1,5)\). The radius is half the diameter length.

Error Analysis and Mastery Check

• Average x with x and y with y; never cross-pair coordinates.
• Put parentheses around negative values before adding.
• Divide each coordinate sum by 2.
• Do not confuse midpoint with distance; midpoint is a point and must be an ordered pair.
• A missing endpoint is not found by averaging the known endpoint and midpoint again; reverse the average.

The midpoint of AB is \((-1,4)\), and A is \((5,-2)\). Then
\[ B=(2(-1)-5,\ 2(4)-(-2))=(-7,10). \]
Average A and B to confirm \((-1,4)\).