Yard Work in Segments

Learning Focus

Use coordinates to find midpoints, lengths, and endpoints of segments.

Big idea

For endpoints $(x_1,y_1)$ and $(x_2,y_2)$:

• Midpoint: $\left(\frac{x_1+x_2}{2},\ \frac{y_1+y_2}{2}\right)$
• Distance: $\sqrt{(x_2-x_1)^2+(y_2-y_1)^2}$
• Missing endpoint: the midpoint is the average, so $x_2=2x_m-x_1$.

Worked example

Midpoint of $A(2,4)$ and $B(10,8)$: $\left(\frac{12}{2},\frac{12}{2}\right)=(6,6)$.

Distance from $(1,2)$ to $(7,10)$: $\sqrt{6^2+8^2}=10$.

If $M(4,3)$ is the midpoint and $A=(1,-1)$, then $B=(7,7)$.

Takeaways

• Midpoint is the average of coordinates.
• Distance is the Pythagorean theorem.
• Work backward for a missing endpoint.