Quadrilaterals: Basic Concepts and Properties

Try this

Click “Show: Angles”. Drag anything: the four angles always add up to $36 0^{\circ}$ . Drag a vertex inside the line joining its two neighbors and the quadrilateral becomes concave: one interior angle is now more than $18 0^{\circ}$ , and the sum is still $36 0^{\circ}$ .
The green quadrilateral joins the midpoints of the four sides. However irregular the original quadrilateral is, it is always a parallelogram, and the reading shows its area is always half the area of the original.
Click “Show: Diagonals” and drag the vertices again: each side of the green quadrilateral stays parallel to one of the diagonals.
Use “Snap to:” to turn the quadrilateral into a trapezoid, a parallelogram, a rectangle, a rhombus and a square in turn, and watch the labels in the family tree light up level by level: the square lights up “parallelogram”, “rectangle” and “rhombus” all at once.

Predict first

Join the midpoints of the four sides of any convex quadrilateral in order. What fraction of the original area does the new quadrilateral have?

\frac{1}{4}

\frac{1}{3}

\frac{1}{2}

It depends on the shape

Choose an answer first; then you can see the result.

Why: a diagonal turns a quadrilateral into triangles

The most basic way to study a quadrilateral is to draw a diagonal, cut it into two triangles, and use what you know about triangles.

Why the midpoint quadrilateral is always a parallelogram

Step 1 / 5

Four midpoints
Let $E$ , $F$ , $G$ , $H$ be the midpoints of $A B$ , $B C$ , $C D$ , $D A$ . We want to prove that $E F G H$ is a parallelogram. There are no useful theorems about the quadrilateral itself…
Draw the diagonal AC
The same on the other side
One pair of opposite sides parallel and equal
Concave works too

The same idea gives the angle sum: a diagonal splits the quadrilateral into two triangles, so the angles add up to $2 \times 18 0^{\circ} = 36 0^{\circ}$ . So the four angles of a quadrilateral cannot all be acute (the sum would be less than $36 0^{\circ}$ ), and they cannot all be obtuse either.

Remove one condition

Break it

Counterexample lab: remove one condition

Conclusion: The midpoint quadrilateral is a rectangle

In this quadrilateral the diagonal $A C$ is horizontal and $B D$ is vertical. Drag $A$ and $C$ left and right along the grid, and $B$ and $D$ up and down (keeping them perpendicular), and watch the “Midpoint quad is a” reading.

Practice

Exercise 1Basic

Three angles of a quadrilateral are $8 0^{\circ}$ , $9 5^{\circ}$ and $11 0^{\circ}$ . Find the fourth angle.

Exercise 2Basic

The four angles of a quadrilateral are in the ratio $1 : 2 : 3 : 4$ . How many degrees is the largest angle?

Exercise 3Advanced

Quadrilateral $A B C D$ has diagonals $A C = 10$ and $B D = 6$ . What is the perimeter of the quadrilateral formed by joining the midpoints of its sides in order?

Exercise 4Challenge

The diagonals of quadrilateral $A B C D$ are perpendicular, with $A C = 8$ and $B D = 6$ . Find the area of its midpoint quadrilateral.

Try this

Why: a diagonal turns a quadrilateral into triangles

Why the midpoint quadrilateral is always a parallelogram

Remove one condition

Counterexample lab: remove one condition

Practice

Mistake Museum · Exhibits from this stop

Judging the midpoint quadrilateral by the sides of the original

Thinking a concave quadrilateral has an angle sum other than 360°