Lesson 4: The Midpoint Formula

Learning Objectives

By the end of this lesson, students will be able to:

  • Apply the midpoint formula to find the center point of a line segment
  • Understand midpoint as the average of coordinates
  • Use midpoints to solve geometric problems
  • Find unknown endpoints when given a midpoint
  • Apply midpoint concepts to real-world situations

Introduction

The midpoint of a line segment is the point that divides it into two equal parts - exactly halfway between the two endpoints. This concept is crucial in many applications: finding the center of a bridge, determining meeting points, locating optimal facility placements, and solving geometric problems.

The Midpoint Formula

For a line segment with endpoints \((x_1, y_1)\) and \((x_2, y_2)\), the midpoint M is:

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

Understanding the Formula

The midpoint formula simply takes the average of the x-coordinates and the average of the y-coordinates: - Midpoint x-coordinate: \(\frac{x_1 + x_2}{2}\) - Midpoint y-coordinate: \(\frac{y_1 + y_2}{2}\)

Think of it as finding the “balance point” between two locations.

Basic Examples

Example 1: Simple Midpoint Calculation

Find the midpoint of the line segment joining A(2, 3) and B(8, 7).

Solution: \[ M = \left(\frac{2 + 8}{2}, \frac{3 + 7}{2}\right) \] \[ M = \left(\frac{10}{2}, \frac{10}{2}\right) \] \[ M = (5, 5) \]

The midpoint is at (5, 5).

Example 2: Midpoint with Negative Coordinates

Find the midpoint of the line segment joining P(-4, 6) and Q(2, -2).

Solution: \[ M = \left(\frac{-4 + 2}{2}, \frac{6 + (-2)}{2}\right) \] \[ M = \left(\frac{-2}{2}, \frac{4}{2}\right) \] \[ M = (-1, 2) \]

Example 3: Midpoint with the Origin

Find the midpoint of the line segment joining O(0, 0) and R(10, 6).

Solution: \[M = \left(\frac{0 + 10}{2}, \frac{0 + 6}{2}\right)\] \[M = (5, 3)\]

Note: The midpoint between the origin and any point \((a, b)\) is \(\left(\frac{a}{2}, \frac{b}{2}\right)\).

Example 4: Midpoint with Fractional Coordinates

Find the midpoint of S(1.5, 2.5) and T(4.5, 6.5).

Solution: \[M = \left(\frac{1.5 + 4.5}{2}, \frac{2.5 + 6.5}{2}\right)\] \[M = \left(\frac{6}{2}, \frac{9}{2}\right)\] \[M = (3, 4.5)\]

Real-World Applications

Example 5: Meeting Point

Two friends live at coordinates A(2, 8) and B(10, 4). They want to meet exactly halfway between their houses. Where should they meet?

Solution: \[M = \left(\frac{2 + 10}{2}, \frac{8 + 4}{2}\right) = (6, 6)\]

They should meet at location (6, 6).

Example 6: Facility Placement

A straight road connects town A at coordinates (10, 20) to town B at coordinates (50, 60). The council wants to place a rest stop at the midpoint. What are the coordinates?

Solution: \[M = \left(\frac{10 + 50}{2}, \frac{20 + 60}{2}\right) = (30, 40)\]

The rest stop should be built at (30, 40).

Example 7: Bridge Center Point

A bridge spans across a river from point P(-12, 5) to point Q(8, 15). Engineers need to place a support at the center of the bridge. What are its coordinates?

Solution: \[M = \left(\frac{-12 + 8}{2}, \frac{5 + 15}{2}\right)\] \[M = \left(\frac{-4}{2}, \frac{20}{2}\right)\] \[M = (-2, 10)\]

The support should be at (-2, 10).

Example 8: Emergency Services

An ambulance station needs to be equidistant from two hospitals: - Hospital A: (5, 12) - Hospital B: (15, 8)

Where should the station be located to be exactly halfway between them?

Solution: \[M = \left(\frac{5 + 15}{2}, \frac{12 + 8}{2}\right) = (10, 10)\]

The station should be at (10, 10).

Working Backwards: Finding Unknown Endpoints

Sometimes we know the midpoint and one endpoint, and need to find the other endpoint.

Example 9: Finding the Other Endpoint

The midpoint of segment AB is M(5, 7). If point A is at (2, 3), find the coordinates of point B.

Solution:

Let B = \((x, y)\)

Using the midpoint formula: \[\frac{2 + x}{2} = 5 \text{ and } \frac{3 + y}{2} = 7\]

Solve for x: \[2 + x = 10\] \[x = 8\]

Solve for y: \[3 + y = 14\] \[y = 11\]

Therefore, B = (8, 11)

Check: Midpoint of (2, 3) and (8, 11) = \(\left(\frac{2+8}{2}, \frac{3+11}{2}\right) = (5, 7)\) ✓

Example 10: Finding the Starting Point

A delivery driver ends their route at point E(20, 30). The midpoint of their route was at M(12, 18). Where did they start?

Solution:

Let the starting point be S\((x, y)\)

\[\frac{x + 20}{2} = 12 \text{ and } \frac{y + 30}{2} = 18\]

Solve for x: \[x + 20 = 24\] \[x = 4\]

Solve for y: \[y + 30 = 36\] \[y = 6\]

The driver started at (4, 6).

Example 11: Symmetric Point

Point P(3, 5) is given. Find the point Q that is symmetric to P with respect to the origin (0, 0).

Solution:

If the origin is the midpoint between P and Q, then:

Let Q = \((x, y)\)

\[\frac{3 + x}{2} = 0 \text{ and } \frac{5 + y}{2} = 0\]

\[x = -3 \text{ and } y = -5\]

Point Q is at (-3, -5).

This is reflection through the origin.

Multiple Midpoints

Example 12: Trisecting a Line Segment

Points A(0, 0) and B(9, 6) are the endpoints of a segment. Find two points that divide the segment into three equal parts.

Solution:

First, find the midpoint M of AB: \[M = \left(\frac{0+9}{2}, \frac{0+6}{2}\right) = (4.5, 3)\]

Now find the midpoint of AM: \[P = \left(\frac{0+4.5}{2}, \frac{0+3}{2}\right) = (2.25, 1.5)\]

And the midpoint of MB: \[Q = \left(\frac{4.5+9}{2}, \frac{3+6}{2}\right) = (6.75, 4.5)\]

However, this doesn’t trisect. To trisect, we need: - First point at 1/3 of the way: \((x_1 + \frac{1}{3}(x_2-x_1), y_1 + \frac{1}{3}(y_2-y_1)) = (3, 2)\) - Second point at 2/3 of the way: \((x_1 + \frac{2}{3}(x_2-x_1), y_1 + \frac{2}{3}(y_2-y_1)) = (6, 4)\)

The two trisection points are (3, 2) and (6, 4).

Example 13: Median of a Triangle

A triangle has vertices at A(0, 0), B(6, 0), and C(3, 6). Find the median from vertex A to side BC.

Solution:

Step 1: Find the midpoint M of BC: \[M = \left(\frac{6+3}{2}, \frac{0+6}{2}\right) = (4.5, 3)\]

Step 2: The median is the line segment from A(0, 0) to M(4.5, 3).

The median from A has endpoint at (4.5, 3).

Geometric Properties Using Midpoints

Example 14: Proving a Parallelogram

Show that quadrilateral ABCD with vertices A(1, 2), B(5, 3), C(6, 6), and D(2, 5) is a parallelogram by showing that the diagonals bisect each other.

Solution:

Find the midpoint of diagonal AC: \[M_{AC} = \left(\frac{1+6}{2}, \frac{2+6}{2}\right) = (3.5, 4)\]

Find the midpoint of diagonal BD: \[M_{BD} = \left(\frac{5+2}{2}, \frac{3+5}{2}\right) = (3.5, 4)\]

Since both diagonals have the same midpoint, they bisect each other, proving ABCD is a parallelogram.

Example 15: Finding the Center of a Circle

Three points on a circle are A(0, 4), B(4, 0), and C(0, -4). The center lies on the perpendicular bisector of any chord. Find an approximation of the center.

Solution:

Find midpoint of AB: \[M_{AB} = \left(\frac{0+4}{2}, \frac{4+0}{2}\right) = (2, 2)\]

The center is somewhere on the perpendicular bisector through (2, 2).

Find midpoint of AC: \[M_{AC} = \left(\frac{0+0}{2}, \frac{4+(-4)}{2}\right) = (0, 0)\]

This suggests the center might be near the origin. By symmetry with points A and C, the center is at (0, 0) and the radius is 4.

Connection to Other Lessons

Practice Exercises

Worksheet 1C: Midpoint Practice

  1. Find the midpoint of the line segment from (2, 8) to (10, 4).
  2. Find the midpoint of the line segment from (-3, 7) to (5, -1).

Set 1: Basic Midpoint Calculations

  1. Find the midpoint of the line segment joining:
    1. A(4, 6) and B(10, 12)
    2. C(-2, 5) and D(6, -3)
    3. E(0, 8) and F(8, 0)
    4. G(-5, -7) and H(-1, -3)
  2. Find the midpoint between:
    1. The origin and (12, 16)
    2. (3.5, 7.2) and (8.5, 4.8)
    3. \((-\frac{1}{2}, 3)\) and \((\frac{5}{2}, 7)\)

Set 2: Finding Unknown Endpoints

  1. The midpoint of segment PQ is M(6, 4). If P is at (2, 1), find Q.

  2. The midpoint of segment RS is M(-3, 5). If S is at (1, 9), find R.

  3. Point A(5, 8) is one endpoint. The midpoint is M(3, 4). Find the other endpoint B.

  4. A segment has midpoint (0, 0) and one endpoint at (7, -3). Find the other endpoint.

Set 3: Applications

  1. Two warehouses are located at A(12, 18) and B(28, 34). A distribution center will be built exactly halfway between them. What are its coordinates?

  2. A park bench should be placed at the midpoint between two trees at positions T₁(-8, 6) and T₂(4, 14). Where should the bench go?

  3. On a map, a school is at (15, 20) and a library is at (35, 40). Where should a crosswalk be placed on a straight path connecting them, exactly at the midpoint?

Set 4: Geometric Applications

  1. A rectangle has opposite vertices at (2, 3) and (10, 11). Find the coordinates of the center of the rectangle.

  2. Triangle ABC has vertices A(0, 0), B(8, 0), and C(4, 6). Find:

    1. The midpoint of side AB
    2. The midpoint of side BC
    3. The midpoint of side AC

  3. Show that the quadrilateral with vertices P(1, 1), Q(5, 2), R(6, 6), and S(2, 5) is a parallelogram by proving the diagonals bisect each other.

Set 5: Challenge Problems

  1. Points A, B, and C are collinear (on the same line) with B between A and C. If A is at (2, 3), B is at (5, 7), and B is the midpoint of AC, find C.

  2. The midpoint of XY is M(4, 6) and the midpoint of YZ is N(7, 9). If X is at (1, 3), find Z.

  3. Three vertices of a parallelogram are A(1, 2), B(5, 4), and C(7, 7). Find all possible locations for the fourth vertex D. (Hint: There are three possible parallelograms)

  4. A segment AB is divided into four equal parts by points P, Q, and R (in that order). If A is at (0, 0) and B is at (12, 8), find the coordinates of P, Q, and R.

Real-World Applications

Urban Planning

City planners use midpoints to: - Determine optimal locations for public facilities - Place street lights evenly along roads - Design bus stops at convenient intervals

Sports

  • Finding the center circle of a sports field
  • Placing mid-court lines in basketball
  • Determining fair meeting points for tournaments

Engineering

  • Locating support beams at the center of bridges
  • Finding balance points in structural design
  • Calculating center of mass for symmetric objects

Additional Resources

Interactive Tools

  • Desmos: Plot segments and automatically show midpoints
  • GeoGebra: Explore midpoint properties dynamically
  • Online midpoint calculators for checking work

Extension Topics

  • Weighted Midpoints: When one endpoint is “heavier” than the other
  • Centroids: The “average” point of a triangle (uses midpoints)
  • Perpendicular Bisectors: Lines through midpoints perpendicular to segments

Key Formulas Summary

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

Finding Other Endpoint: If M\((m_x, m_y)\) is the midpoint of AB and A\((x_1, y_1)\) is known: \[B = (2m_x - x_1, 2m_y - y_1)\]

Verification: Distance from A to M equals distance from M to B.

Homework

Complete Practice Exercises Sets 1 and 2, and at least 3 problems from Set 3.

For additional worksheets and practice problems, visit the Resources section.

  1. (6, 6)
  2. (1, 3)

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