MPM2D Midpoint and Length of a Line Segment: Notes and Practice

Amara and Ben live in the same town. On a city map drawn on a grid where each square is $1$ km wide, Amara's home is at $A (- 2, 1)$ and Ben's is at $B (4, 9)$ . They want to meet at a café exactly halfway between their homes, and they want to know how far apart they live in a straight line ("as the crow flies"). You could measure the map with a ruler, but a ruler gives an estimate that depends on how carefully the map was drawn. Analytic geometry answers both questions exactly, using nothing but the coordinates. By the end of this lesson you will be able to say that the café belongs at $(1, 5)$ and that the homes are exactly $10$ km apart, and you will know why.

A line segment is the part of a line between two endpoints. It is named by its endpoints, so the segment from $A$ to $B$ is written $A B$ . The midpoint of a segment is the point on the segment that is the same distance from both endpoints; it cuts the segment into two equal halves. The length of a segment is the distance between its endpoints. To write formulas that work for any segment, label the endpoints with subscripts: the first endpoint is $(x_{1}, y_{1})$ and the second is $(x_{2}, y_{2})$ . The small $1$ and $2$ are labels, not exponents: $x_{1}$ is read "x one" and just means "the x-coordinate of the first point".

Fig 1The segment from A(−2, 1) to B(4, 9) on a grid. Its midpoint M(1, 5) is halfway across and halfway up. A right triangle with legs 6 and 8 has the segment as its hypotenuse, so the length is 10.

Wide aerial view of downtown Toronto from the CN Tower, with office towers in front and city blocks stretching north to the horizon. — PhotoToronto's streets are laid out in blocks that work like a coordinate grid, so the point halfway between two places, and the straight-line distance between them, can be calculated from coordinates.DXR, CC BY-SA 4.0, via Wikimedia Commons

Developing the midpoint formula

Look only at the x-coordinates first. Going from $A$ to $B$ , the x-coordinate runs from $- 2$ to $4$ . The number halfway between $- 2$ and $4$ on a number line is their average (mean): $\frac{- 2 + 4}{2} = \frac{2}{2} = 1$ . Now the y-coordinates: halfway between $1$ and $9$ is $\frac{1 + 9}{2} = \frac{10}{2} = 5$ . So the midpoint is $M (1, 5)$ . Why can the two coordinates be treated separately? From $A$ to $B$ the segment goes $6$ right (the run) and $8$ up (the rise). Starting at $A$ and going half the run and half the rise ( $3$ right and $4$ up) lands on the segment, because the slope stays $\frac{4}{3} = \frac{8}{6}$ . That point is exactly halfway along. The small triangle from $A$ to $M$ is a half-size copy of the big triangle from $A$ to $B$ .

M = (\frac{x _{1} + x _{2}}{2}, \frac{y _{1} + y _{2}}{2})

Midpoint formula: average the x-coordinates, then average the y-coordinates.

Quick check for a midpoint

The midpoint's x-coordinate must lie between the two endpoint x-coordinates, and the same for y. For

A (- 2, 1)

and

B (4, 9)

1

is between

- 2

and

4

, and

5

is between

1

and

9

. If your answer is outside those ranges, you probably subtracted instead of adding.

The formula also runs backwards. Suppose $M (2, - 1)$ is the midpoint of $A B$ and you know $A (- 3, 4)$ but not $B$ . Getting from $A$ to $M$ takes $5$ right (from $- 3$ to $2$ ) and $5$ down (from $4$ to $- 1$ ). Because $M$ is halfway, getting from $M$ to $B$ takes the same move again: $5$ right and $5$ down, so $B = (2 + 5, - 1 - 5) = (7, - 6)$ . Algebraically, set $\frac{- 3 + x _{2}}{2} = 2$ , so $- 3 + x_{2} = 4$ and $x_{2} = 7$ ; and set $\frac{4 + y _{2}}{2} = - 1$ , so $4 + y_{2} = - 2$ and $y_{2} = - 6$ . Either way, check by finding the midpoint of $(- 3, 4)$ and $(7, - 6)$ : it is $(\frac{4}{2}, \frac{- 2}{2}) = (2, - 1)$ .

VideoThe midpoint formula comes from averaging the x-coordinates and the y-coordinates of the two endpoints.Watch on YouTube

Developing the length formula

Horizontal and vertical segments are easy: subtract. From $(1, 3)$ to $(7, 3)$ is $7 - 1 = 6$ units, and from $(2, - 4)$ to $(2, 5)$ is $5 - (- 4) = 9$ units. A slanted segment needs one more idea. Draw a horizontal line through $A (- 2, 1)$ and a vertical line through $B (4, 9)$ . They meet at $C (4, 1)$ , the corner of a right triangle whose hypotenuse is $A B$ . The horizontal leg $A C$ has length $4 - (- 2) = 6$ and the vertical leg $CB$ has length $9 - 1 = 8$ . The Pythagorean theorem ( $c^{2} = a^{2} + b^{2}$ , true for every right triangle) gives $A B^{2} = 6^{2} + 8^{2} = 36 + 64 = 100$ , so $A B = 100 = 10$ . Amara and Ben live $10$ km apart. Replacing the numbers by the general coordinates gives the length formula (also called the distance formula):

d = (x_{2} - x_{1})^{2} + (y_{2} - y_{1})^{2}

Length of the segment from (x1, y1) to (x2, y2): the Pythagorean theorem with the run and rise as the legs.

Three details make the formula reliable. First, the order of the points does not matter: $(4 - (- 2))^{2} = 6^{2} = 36$ and $(- 2 - 4)^{2} = (- 6)^{2} = 36$ , because squaring removes the sign. Second, a difference can be negative, so put it in brackets before squaring: $(- 6)^{2} = 36$ , but a calculator reads $- 6^{2}$ as $- 36$ . Third, most lengths are not whole numbers. For $P (- 1, 3)$ and $Q (5, - 1)$ : $x_{2} - x_{1} = 5 - (- 1) = 6$ and $y_{2} - y_{1} = - 1 - 3 = - 4$ , so $d = 36 + 16 = 52$ . Give the exact answer as a radical in simplest form by taking out the largest perfect-square factor, $52 = 4 \times 13 = 213$ , and give a decimal approximation when one is asked for: $213 \approx 7.2$ .

Endpoints	Run	Rise	Sum of squares	Exact length	Decimal
$(- 2, 1)$ and $(4, 9)$	$6$	$8$	$36 + 64 = 100$	$10$	$10.0$
$(- 1, 3)$ and $(5, - 1)$	$6$	$- 4$	$36 + 16 = 52$	$213$	$7.2$
$(0, 0)$ and $(3, 3)$	$3$	$3$	$9 + 9 = 18$	$32$	$4.2$
$(1, - 2)$ and $(- 3, 0)$	$- 4$	$2$	$16 + 4 = 20$	$25$	$4.5$

Run is x2 minus x1 and rise is y2 minus y1. Decimals are rounded to one place.

Loading interactive…

Drag the endpoints of a segment on the grid and watch its exact midpoint, length and slope update, with the working shown for each.

The root of a sum is not a sum of roots

36 + 64 = 100 = 10

, but

36 + 64 = 6 + 8 = 14

. Add the squares first, then take one square root of the total. Adding the roots measures the walk across and then up, which is longer than the straight path for any slanted segment.

With both formulas you can measure any shape whose corners have coordinates. The perimeter of a triangle is the sum of its three side lengths, each found with the length formula. A side length that is a radical can stay exact until the end: a triangle with sides $10$ , $5$ and $97$ has perimeter $15 + 97 \approx 24.8$ . Round only the final answer, not the pieces along the way, so rounding errors do not build up.

Midpoint and Length of a Line Segment

Key things to remember

Developing the midpoint formula

Developing the length formula

Worked examples

Where marks get lost

Flashcards 8

Practice