Prove It with Algebra

Learning Focus

Use slope, distance, and midpoint formulas to prove geometric claims in the coordinate plane.

Big idea

In a coordinate proof, place figures on a grid and use algebra.

• Parallel sides: equal slopes.
• Perpendicular sides (right angle): slopes multiply to $-1$.
• Equal lengths: distance formula.
• Parallelogram: diagonals share a midpoint (or both pairs of opposite sides parallel).

Worked example

Quadrilateral $A(0,0),B(4,1),C(6,5),D(2,4)$.

• $AB$ shifts $(4,1)$ and $DC$ shifts $(4,1)$, so $AB\parallel DC$ and they are equal.
• Midpoint of $\overline{AC}$: $(3,2.5)$; midpoint of $\overline{BD}$: $(3,2.5)$.

The diagonals bisect each other, so $ABCD$ is a parallelogram.

Takeaways

• State what must be shown.
• Compute slopes, lengths, or midpoints.
• Conclude with the theorem.