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Analytic Geometry · Analytic Geometry

The Midpoint of a Line Segment

2.1About 65 minutes
Learning goals
  • I can explain why the midpoint's coordinates are the averages of the endpoints' coordinates.
  • I can use the midpoint formula to find the midpoint of a segment.
  • I can find the midpoints of the sides of a triangle and verify them concretely.
  • I can find a missing endpoint when I know the midpoint and the other endpoint.

Words to know

line segment
The part of a line between two endpoints.
midpoint
The point on a segment that is the same distance from both endpoints.
bisect
To cut into two equal parts.
analytic geometry
Geometry done with coordinates and algebra (also called coordinate geometry).
congruent
Having exactly the same size and shape.

Meeting halfway

Two friends live in different parts of town and want to meet at a café exactly halfway between their homes. On a city map with a grid, where should the café be? The answer is the midpoint of the line segment joining their homes. In this lesson you will develop a formula for the midpoint and use it to solve problems.

Remember from Grade 9

In Grade 9 you used the midpoint of a chord when you learned that the perpendicular bisector of a chord passes through the centre of a circle, and that the circumcentre of a triangle is where the perpendicular bisectors of its sides meet. Now you will find midpoints exactly from coordinates.

Halfway on a number line

Start in one dimension. What number is halfway between −3 and 7? The distance between them is 7 − (−3) = 10, so halfway is 5 steps from each end: −3 + 5 = 2. Notice that 2 is the average, or mean, of the two numbers: (−3 + 7) ÷ 2 = 4 ÷ 2 = 2.

-4-3-2-1012345678+5+5
The number halfway between −3 and 7 is their average, 2. It is 5 units from each end.

Developing the midpoint formula

Now take the segment from A(−3, 5) to B(7, −1). Going from A to B, x changes by 7 − (−3) = 10 (the run) and y changes by −1 − 5 = −6 (the rise, here a fall of 6).

The midpoint M is halfway along the segment, so to get from A to M you go half the run and half the rise: right 5 and down 3. That gives M(−3 + 5, 5 − 3) = M(2, 2). The two small right triangles in the diagram are congruent (same size and shape), which is why M is exactly halfway.

The x-coordinate of M is halfway between −3 and 7, which is their average, 2. The y-coordinate is halfway between 5 and −1, which is their average, 2. This works for any segment:

−3−2−112345678−2−11234560xyA(−3, 5)B(7, −1)M(2, 2)run 5run 5fall 3fall 3
From A to M is half the run (5) and half the fall (3); so is M to B. The two dashed right triangles are congruent, so the tick marks show AM = MB. M(2, 2) is the midpoint.

The midpoint formula

The midpoint M of the segment joining A(x₁, y₁) and B(x₂, y₂) is

M = ( (x₁ + x₂)/2 , (y₁ + y₂)/2 )

In words: average the x-coordinates and average the y-coordinates. It does not matter which point you call A and which you call B.

Worked example

Example 1: Find a midpoint

Find the midpoint of the segment joining C(−5, 1) and D(3, 7).

  1. x-coordinate: (−5 + 3)/2 = −2/2 = −1.
  2. y-coordinate: (1 + 7)/2 = 8/2 = 4.
  3. Check by distance along each axis: from −5 to −1 is 4 and from −1 to 3 is 4 ✓; from 1 to 4 is 3 and from 4 to 7 is 3 ✓.

The midpoint is (−1, 4).

Worked example

Example 2: Midpoints of the sides of a triangle

A triangle has vertices P(−4, 2), Q(6, 4) and R(2, −6). Find the midpoint of each side.

  1. Midpoint of PQ: ((−4 + 6)/2, (2 + 4)/2) = (2/2, 6/2) = (1, 3).
  2. Midpoint of QR: ((6 + 2)/2, (4 + (−6))/2) = (8/2, −2/2) = (4, −1).
  3. Midpoint of PR: ((−4 + 2)/2, (2 + (−6))/2) = (−2/2, −4/2) = (−1, −2).
  4. Verify concretely: plot the triangle on graph paper, fold each side so its endpoints meet, and check that the crease crosses the side at your midpoint. Or use free dynamic geometry software such as GeoGebra: its Midpoint tool should give the same three points.

The midpoints are (1, 3), (4, −1) and (−1, −2).

−5−4−3−2−11234567−6−5−4−3−2−1123450xyP(−4, 2)Q(6, 4)R(2, −6)(1, 3)(4, −1)(−1, −2)
Triangle PQR with the midpoint of each side marked by a larger ringed dot. Each midpoint is the average of the two vertices at the ends of that side.
Worked example

Example 3: Find a missing endpoint

M(3, −2) is the midpoint of segment AB. A is (−1, 4). Find B.

  1. Let B = (x, y). Use the formula: (−1 + x)/2 = 3 and (4 + y)/2 = −2.
  2. Solve: −1 + x = 6, so x = 7. And 4 + y = −4, so y = −8.
  3. Another way (steps): from A to M, x goes from −1 to 3 (+4) and y from 4 to −2 (−6). Repeat the same steps from M: x = 3 + 4 = 7, y = −2 − 6 = −8.
  4. Check: ((−1 + 7)/2, (4 + (−8))/2) = (3, −2) ✓.

B(7, −8).

Worked example

Example 4: The centre of a circle

The endpoints of a diameter of a circular skating rink on a park map are (−5, 1) and (3, 7). Where is the centre of the rink?

  1. The centre of a circle is the midpoint of any diameter.
  2. Centre: ((−5 + 3)/2, (1 + 7)/2) = (−1, 4).

The centre is at (−1, 4) (the same segment as Example 1).

Just for fun

Why does the midpoint get invited to every party?

Because it always meets people halfway.

Hands-on activity

Fold to find the middle

You need: a sheet of graph paper (or plain paper with a grid you draw), a ruler and a pencil.

  1. Draw x- and y-axes. Plot two points, for example A(−6, 3) and B(4, −5), and join them with a ruler.
  2. Predict the midpoint with the formula: ((−6 + 4)/2, (3 + (−5))/2) = (−1, −1).
  3. Fold the paper so that A lands exactly on top of B (hold it up to a window to line them up). Crease firmly and unfold.
  4. The crease crosses AB at the midpoint. Does it pass through (−1, −1)?
  5. Bonus: the crease is a special line. Measure the angle where it crosses AB with the corner of a book or index card. What do you notice?

What to look for: the crease should pass through (−1, −1). It also meets AB at a right angle: the crease is the right bisector (perpendicular bisector) of AB, which you will use in the lesson on right bisectors.

Your turn

1

Find the midpoint of (2, 8) and (6, 4).

2

Find the midpoint of (−7, 3) and (1, −5).

3

Find the midpoint of (0, 0) and (−9, 4).

4

Find the midpoint of (3, −2) and (8, 5).

5

M(4, 1) is the midpoint of PQ and P is (1, 6). Find Q.

6

A triangle has vertices A(0, 0), B(8, 2) and C(4, −6). Find the midpoints of its three sides.

7

Two friends live at (−3, 8) and (5, −2) on a town grid (units in blocks). Where is the café halfway between them?

More practice: warm-up

1

What number is halfway between 4 and 18?

2

What number is halfway between −10 and 2?

3

Find the midpoint of (1, 1) and (5, 9).

4

Find the midpoint of (−2, −4) and (2, 4).

5

In words, how do you find the midpoint of a segment?

More practice: on your own

1

Find the midpoint of (−1.5, 4) and (6.5, −3).

2

The midpoint of AB is (−2, 5). A is (3, 9). Find B.

3

A diameter of a circle has endpoints (−4, −3) and (4, 3). Find the centre. What special point is it?

4

Find the point that divides the segment from (0, 0) to (12, −8) into four equal parts (find all three points).

5

Show that the midpoint of (a, b) and (−a, −b) is always the origin.

Challenge

1

The midpoints of the sides of a triangle are (1, 3), (4, −1) and (−1, −2). Find the vertices. (Hint: compare with Example 2.)

2

Point A is (2, 1). Point B is on the line y = 3x. The midpoint of AB lies on the x-axis. Find B.

3

A parallelogram has three vertices A(1, 1), B(6, 2) and C(8, 6), in order. Use the fact that the diagonals of a parallelogram bisect each other to find the fourth vertex D.

Check your understanding

Quiz

1.

What is the midpoint of (−4, 6) and (8, 2)?

2.

The midpoint formula finds each coordinate by:

3.

M(1, −3) is the midpoint of AB, and A is (−2, 0). What is B?

4.

A diameter of a circle has endpoints (2, 7) and (8, −1). Where is the centre?

5.

What is the midpoint of (3, 5) and (4, 10)?

6.

A student finds the midpoint of (−6, 2) and (4, 8) as (5, 3). What mistake did they make?

For parents and teachers

Big idea: the midpoint of a segment has coordinates that are the averages of the endpoints' coordinates. This lesson teaches expectation 2.1 (develop the midpoint formula and use it to solve problems, including the midpoints of the sides of a triangle).

How to help: Ask your teen to explain why the formula works (half the run and half the rise, or "halfway is the average"). Do the folding activity together: it is a concrete check that needs only paper.

Common difficulties:

  • Subtracting the coordinates instead of adding them (subtracting is for slope and length, which come next).
  • Sign errors with negative coordinates: (−7 + 1)/2 = −3, not −4 or 4.
  • Finding a missing endpoint: some students average the midpoint and the known endpoint. Encourage writing the formula with an unknown and solving, or the "repeat the same steps" method.