Midpoint and distance of a coordinate segment

Endpoint A

\(\displaystyle (-7,4)\)

Endpoint B

\(\displaystyle (9,-2)\)

Find the midpoint of the segment

The midpoint is halfway between the endpoints in each coordinate direction

Keep A fixed. Move B from vertical coordinate −6 to 6, checking intermediate positions. Compare the changes in B and midpoint M

When B moves up 4 units, how far does M move?

Endpoint A

\(\displaystyle A=(-2,4)\)

Endpoint B

\(\displaystyle B=(7,-1)\)

Find the horizontal coordinate of the midpoint

Endpoint A

\(\displaystyle A=(-2,4)\)

Endpoint B

\(\displaystyle B=(7,-1)\)

Find the vertical coordinate of the midpoint

Average positions, not just changes

A midpoint has two coordinates. Average the horizontal positions separately from the vertical positions. Fractional coordinates are valid

1. 1Position Half the change is a displacement from A. Add it to A to get the midpoint \(\displaystyle x_M=x_A+\frac{x_B-x_A}{2}\)
2. 2Equivalent average Simplifying gives the coordinate average \(\displaystyle \begin{gathered}x_M=\frac{x_A+x_B}{2}\\y_M=\frac{y_A+y_B}{2}\end{gathered}\)

A=(−3,−2), B=(3,2). The coordinate separations are 6 and 4 units. Which expression gives the straight-line distance?

Move B through vertical coordinates −2, 0, 2, 4 and 6. Compare the dashed legs, squared length and straight-line length. At −2 the segment is horizontal

What happens when the vertical separation becomes zero?

Use Pythagoras on coordinate changes

Straight-line distance is nonnegative. Signed changes can be negative, but their squares are nonnegative. Swapping A and B changes both signs without changing the distance

1. 1Two perpendicular changes Use the same endpoint order for both differences \(\displaystyle \begin{gathered}\Delta x=x_B-x_A\\\Delta y=y_B-y_A\\d=\sqrt{(\Delta x)^2+(\Delta y)^2}\end{gathered}\)
2. 2One changing coordinate A horizontal or vertical segment has one zero change. Its length is the magnitude of the other change

Endpoint A

\(\displaystyle A=(-7,3)\)

Endpoint B

\(\displaystyle B=(4,3)\)

Find the length of this horizontal segment in units

Endpoint A

\(\displaystyle A=(2,-5)\)

Endpoint B

\(\displaystyle B=(2,4)\)

Find the length of this vertical segment in units

Endpoint A

\(\displaystyle A=(-3,-2)\)

Endpoint B

\(\displaystyle B=(3,6)\)

Find the straight-line length in units

Endpoint P

\(\displaystyle (-5,6)\)

Endpoint Q

\(\displaystyle (7,1)\)

Where is the first error in this student's length calculation?

Squared length

\(\displaystyle d^2=52\)

The squared length is 52 square units. Give the length in units to two decimal places

Scale

\(\displaystyle 1\text{ coordinate unit}=2\text{ m}\)

On a plan, A=(−3,−2) and B=(3,2). One coordinate unit represents 2 metres on both axes. Find the actual straight-line distance to two decimal places

Length of each segment

\(\displaystyle \sqrt{2}\text{ m}\)

Total after rounding each length first

\(\displaystyle 10\times1.41=14.10\text{ m}\)

Total using exact lengths

\(\displaystyle 10\sqrt{2}\approx14.14\text{ m}\)

Ten segments have the exact length and totals shown below. What did early rounding do?

Choose the quantity and check the result

A midpoint is an ordered pair. A length is one nonnegative number with length units. Read numerical axis scales rather than measuring the picture

1. 1Midpoint check Its coordinates lie halfway between the corresponding endpoint coordinates
2. 2Length check For an oblique segment the direct distance is longer than either perpendicular separation, but shorter than their sum
3. 3Model check A scaled plan must use the same stated physical meaning on both coordinate axes before applying the ordinary distance formula