Build a frame from two strips

The idea

Whatever the angle between them, if the diagonals of a quadrilateral are equal and bisect each other, it is a rectangle; so a rectangle can also be defined by its diagonals.

Turn it around

We found that in a rectangle the diagonals are equal and bisect each other. Now turn it around. The carpenter takes two equal strips and crosses them at their middles. She ties a thread round the four ends.

At what angle between the strips will the thread make a rectangle?

• Only at 90°
• At any angle
• Only at 60°

The carpenter’s frame

Strip AC is fixed at 8 cm. Change the length of BD, where the strips cross, and the angle between them. Watch the thread.

Equal and bisecting

Whatever the angle between them, if the diagonals of a quadrilateral are equal and bisect each other, it is a rectangle; so a rectangle can also be defined by its diagonals.

Try 60° between the strips. ∆AOB has OA = OB, so its base angles are (180° − 60°) ÷ 2 = 60°. ∆AOD has 120° at O, so its base angles are (180° − 120°) ÷ 2 = 30°.

Each corner is 60° + 30° = 90°. For any angle x it is (90° − x/2) + x/2 = 90°. The matching triangles also make opposite sides equal.

Carpenters in Europe, and farmers in Mozambique building houses, use this method to get a rectangle.

Notes

If the diagonals of a quadrilateral are equal and bisect each other, it is a rectangle, whatever the angle between them.

Check yourself

Two strips of 10 cm cross at their middles at 40°. A thread goes round the four ends. What shape is it?

Which pair of strips gives a rectangle when a thread goes round the ends?

The strips cross at 80°. ∆AOB has OA = OB. Find the angle ∠A = ∠OAB + ∠OAD of the frame.

Answer: 90 °

∠OAB = (180° − 80°) ÷ 2 = 50°. At O the other angle is 100°, so ∠OAD = (180° − 100°) ÷ 2 = 40°. ∠A = 50° + 40° = 90°.

A carpenter makes a rectangle from two equal strips of 12 cm. At what distance from each end should they cross?

Answer: 6 cm

The strips must bisect each other, so each is cut at its middle: 12 ÷ 2 = 6 cm from each end.

• A parallelogram that is not a rectangle. Equal strips that bisect each other always make a rectangle, whatever the angle.
• A rectangle — correct. Yes! The strips are equal and bisect each other, so the angle between them does not matter.
• A rectangle only if the angle is 90°. At 90° it is a square, a special rectangle. At any other angle it is still a rectangle.
• 10 cm and 8 cm, crossing at the middle of both. They bisect each other, but they are not equal. That makes a parallelogram, not a rectangle.
• 10 cm and 10 cm, crossing at the middle of both — correct. Yes! Equal and bisecting.
• 10 cm and 10 cm, crossing at the middle of one strip only. They are equal, but they do not bisect each other. The thread does not make a rectangle.