Trapezoids

Try this

A trapezoid is a quadrilateral with only one pair of parallel opposite sides. The two parallel sides are the bases; the other two sides are the legs.

The green segment $M N$ joins the midpoints of the two legs. It is called the midline of the trapezoid. Drag anything: $M N$ is always half the sum of the two bases, and it looks parallel to them.
Drag $D$ to the left until $D C$ is as long as $A B$ : the “Right now” reading changes from “trapezoid” to “parallelogram”. It is no longer a trapezoid.
Click “Snap to: Isosceles trapezoid”, then turn on “Diagonals”: when the legs are equal, the diagonals are equal too.
Turn on “Translate a leg”: the leg $D A$ slides along the base to position $C E$ . The trapezoid is cut into a parallelogram $A E C D$ and a triangle $C E B$ .

Predict first

A trapezoid has an upper base of $4$ and a lower base of $10$ . How long is its midline?

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

Why: translate, and take the trapezoid apart

The two legs of a trapezoid are neither parallel nor equal, which makes it awkward to study directly. The usual trick is to translate one segment so that the scattered information ends up in one triangle.

Translate a leg: the base angles of an isosceles trapezoid are equal

Step 1 / 5

An isosceles trapezoid
$A B ∥ D C$ and $A D = B C$ . We want to prove $∠ A = ∠ B$ . The two angles are at opposite ends of the lower base, and the two legs do not touch.
Through C, draw a line parallel to DA
Both legs are now in one triangle
Corresponding angles
It works without equal legs too

The midline theorem: MN ∥ AB, MN = (AB + DC) ÷ 2

Step 1 / 4

The midline
$M$ and $N$ are the midpoints of the legs $A D$ and $B C$ . There is no ready-made theorem for trapezoids, but triangles have the midline theorem. Find a way to make $M N$ the midline of some triangle.
Join DN and extend it
The upper base is “turned” next to the lower base
MN is the midline of △DAG

So the area of a trapezoid $= \frac{1}{2} (upper base + lower base) \times height = midline \times height$ .

Remove one condition

Break it

Counterexample lab: remove one condition

Conclusion: The segment joining the midpoints of the legs is $M N = \frac{1}{2} (A B + D C)$

A trapezoid: the two readings are always equal.

Practice

Exercise 1Basic

A trapezoid has an upper base of $5$ and a lower base of $11$ . Find the length of its midline.

Exercise 2Basic

A trapezoid has midline $7$ and height $6$ . Find its area.

Exercise 3Advanced

An isosceles trapezoid has upper base $2$ , lower base $14$ and legs of length $10$ . Find its area.

Exercise 4Challenge

In trapezoid $A B C D$ , $A B ∥ D C$ , the diagonals are perpendicular, $A C ⊥ B D$ , and $A C = 6$ , $B D = 8$ . Find the sum of the bases $A B + D C$ .

Try this

Why: translate, and take the trapezoid apart

Translate a leg: the base angles of an isosceles trapezoid are equal

The midline theorem: MN ∥ AB, MN = (AB + DC) ÷ 2

Remove one condition

Counterexample lab: remove one condition

Practice

Mistake Museum · Exhibits from this stop

Taking “half the difference of the bases” as the midline

Counting a parallelogram as a trapezoid