Midpoint, endpoint 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

Move B upwards through several slider positions, then move it downwards. Compare how far B and its midpoint M move vertically

For the positions you compared, how far did M move when B moved?

Use the coordinates for the quantity you need

A midpoint is a position. A segment length is a distance. To recover an endpoint, continue the same move from the known endpoint through the midpoint

1. 1Midpoint Average matching coordinates separately. The horizontal coordinate comes first \(\displaystyle M=\left(\frac{x_A+x_B}{2},\frac{y_A+y_B}{2}\right)\)
2. 2Length The horizontal and vertical changes form perpendicular sides. Apply Pythagoras and take the positive square root. For a horizontal or vertical segment, use the distance along its single changing coordinate \(\displaystyle \begin{aligned}\Delta x&=x_B-x_A\\\Delta y&=y_B-y_A\\d^2&=(\Delta x)^2+(\Delta y)^2\\d&=\sqrt{(\Delta x)^2+(\Delta y)^2}\end{aligned}\)
3. 3Missing endpoint Double each midpoint coordinate, then subtract the matching coordinate of the known endpoint. These rules show the case where A is known \(\displaystyle \begin{aligned}x_B&=2x_M-x_A\\y_B&=2y_M-y_A\end{aligned}\)
4. 4Exact or rounded A square root can be an exact answer even when it is irrational. If a decimal answer is requested, keep the exact squared length and round the final root once. The example below rounds to two decimal places \(\displaystyle \begin{aligned}d^2&=61\\d&=\sqrt{61}\\d&\approx7.81\text{ units}\end{aligned}\)

Endpoint P

\(\displaystyle (-8,-3)\)

Endpoint Q

\(\displaystyle (7,5)\)

Calculate the length of the segment in units

The horizontal and vertical changes are the perpendicular sides of a right-angled triangle

Midpoint M

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

Endpoint A

\(\displaystyle (4,-1)\)

Which point is the missing endpoint B?

Make the same move from A to M again to reach B

Check the two applications

Use the original coordinates to check each result. The two examples use different operations

1. 1Calculate the length Find both changes. They are perpendicular, so use Pythagoras and take the positive root \(\displaystyle \begin{aligned}P&=(-8,-3),\quad Q=(7,5)\\\Delta x&=7-(-8)=15\\\Delta y&=5-(-3)=8\\d^2&=15^2+8^2\\d^2&=225+64\\d^2&=289\\d&=\sqrt{289}=17\text{ units}\end{aligned}\)
2. 2Recover the endpoint Double the midpoint coordinates and subtract the matching coordinates of A \(\displaystyle \begin{aligned}M&=(-2,3),\quad A=(4,-1)\\x_B&=2(-2)-4=-8\\y_B&=2(3)-(-1)=7\\B&=(-8,7)\end{aligned}\)
3. 3Check the midpoint Average the known and recovered endpoints. Both averages match the supplied midpoint \(\displaystyle \begin{aligned}x_M&=\frac{4+(-8)}{2}=-2\\y_M&=\frac{-1+7}{2}=3\end{aligned}\)

Endpoint P

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

Endpoint Q

\(\displaystyle (7,1)\)

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