Geometry / Unit 7: Polygons and Quadrilaterals
7.6 Quadrilaterals on the Coordinate Plane
Notes
Use formulas to classify a quadrilateral from its vertices.
- Slope checks for parallel sides (equal slopes) and right angles (product $-1$).
- Distance checks for congruent sides and congruent diagonals.
- Midpoint checks whether diagonals bisect each other (same midpoint).
- Example: if the diagonals share a midpoint and are equal in length, the figure is a rectangle.
Practice
The diagonals of a quadrilateral have the same midpoint. What is the quadrilateral at least?
Diagonals that bisect each other prove a parallelogram.
Quadrilateral $ABCD$ has $A(0, 0)$, $B(4, 0)$, $C(4, 3)$ and $D(0, 3)$. What is the length of diagonal $\overline{AC}$?
$\displaystyle \sqrt{4^2 + 3^2} = 5$.
The diagonals of parallelogram $PQRS$ have slopes 2 and $\displaystyle -\frac{1}{2}$. What must $PQRS$ be?
The slopes multiply to $-1$, so the diagonals are perpendicular, which makes the parallelogram a rhombus. Nothing shows a right angle, so it need not be a square.
Ready for more? Take the Unit 7 test.