Squares

Try this

A square is both a rectangle and a rhombus: four equal sides, four right angles, and diagonals that are equal, perpendicular and bisect each other. Rotated $9 0^{\circ}$ about its center, it fits onto itself; rotated $9 0^{\circ}$ about a vertex, one side lands exactly on the neighboring side.

Drag $E$ and $F$ . In general $E F$ and $B E + D F$ are not equal.
Click “Move F so that ∠EAF = 45°”: the two readings are now equal. Drag $E$ somewhere else and click again: still equal.
Now look at the last reading: the perimeter of $△ C E F$ is exactly $10$ , twice the side of the square, wherever $E$ is.
Drag the rotation slider to $9 0^{\circ}$ : the blue $△ A D F$ turns about $A$ into the position of $△ A B G$ . $D F$ becomes $B G$ and joins $B E$ to form one segment $GE$ .

Predict first

Square $A B C D$ has side $5$ . $E$ is on $B C$ , $F$ is on $C D$ , and $∠ E A F = 4 5^{\circ}$ . What is the perimeter of $△ C E F$ ?

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

Why: turn 90° to join the two pieces

If ∠EAF = 45°, then EF = BE + DF

Step 1 / 5

The two segments are apart
We want to prove $E F = B E + D F$ . $B E$ (orange) and $D F$ (blue) are on two different sides of the square, so we cannot add them directly.

$E F = ? B E + D F$
Rotate △ADF clockwise about A
Turn the full 90°
Find the congruent triangles
Read off the conclusion

The key to this method (often called the “half-angle model”): two equal segments with a common endpoint (here $A D = A B$ ) are a signal to rotate, and the angle of rotation is the angle between them. A square is full of such segments, which is why square problems are so well suited to rotation.

Remove one condition

Break it

Counterexample lab: remove one condition

Conclusion: $E F = B E + D F$

$F$ follows $E$ automatically and keeps $∠ E A F = 4 5^{\circ}$ . Drag the rotation slider to $9 0^{\circ}$ , then drag $E$ : $∠ G A E$ stays $4 5^{\circ}$ as well.

Practice

Exercise 1Basic

A square has a diagonal of length $6$ . Find its area.

Exercise 2Advanced

Square $A B C D$ has side $4$ . $E$ is on $B C$ with $B E = 1$ ; $F$ is on $C D$ with $∠ E A F = 4 5^{\circ}$ . Find the length of $E F$ .

Exercise 3Advanced

Inside square $A B C D$ , build an equilateral triangle $A B E$ . Find $∠ D E C$ in degrees.

Exercise 4Challenge

Square $A B C D$ has side $6$ . $E$ is on $B C$ with $B E = 2$ ; $F$ is on $C D$ with $∠ E A F = 4 5^{\circ}$ . Find the area of $△ A E F$ .

Try this

Why: turn 90° to join the two pieces

If ∠EAF = 45°, then EF = BE + DF

Remove one condition

Counterexample lab: remove one condition

Practice

Mistake Museum · Exhibits from this stop

Four equal sides make a square

Perpendicular, equal diagonals make a square