Diagonals of a square

The idea

The diagonals of a square are equal and bisect each other at right angles; diagonals drawn this way always give a square.

Which angle makes a square?

The carpenter has two equal strips of 8 cm. She lays them so that they cross at their middle points. The thread round their ends makes a rectangle, whatever the angle. But she wants a square: all sides equal.

At which angle should the strips cross to make a square?

• 30°
• 60°
• 90°
• 120°

Turn the strips

The strips cross at their middles. Turn the strip BD with the slider. Watch the four sides of the thread: are they equal?

A right angle in the middle

The diagonals of a square are equal and bisect each other at right angles; diagonals drawn this way always give a square.

Why 90°? In a square, OA = OC (the diagonals bisect each other), OB is common, and AB = CB. So ∆BOA ≅ ∆BOC by SSS, and ∠BOA = ∠BOC.

These two angles lie on a straight line, so ∠BOA + ∠BOC = 180°. Equal angles that add up to 180° are 90° each.

Notes

The diagonals of a square are equal and bisect each other at right angles; diagonals drawn this way always give a square.

Check yourself

Which pair of strips makes a square when a thread goes round their ends?

ABCD is a square whose diagonals meet at O. The diagonal AC = 14 cm. How long is OB?

Answer: 7 cm

The diagonals of a square are equal, so BD = AC = 14 cm. They bisect each other, so OB = 14 ÷ 2 = 7 cm.

Two strips of 10 cm cross at their middle points, but at 60° to each other. What does the thread make?

In square ABCD with diagonals meeting at O, why is ∆BOA ≅ ∆BOC?

• Equal strips crossing at their middles at 70°. Equal strips that bisect each other make a rectangle, but at 70° the sides are not all equal.
• Unequal strips crossing at their middles at 90°. At 90° all four sides are equal, but the angles of the shape are not all 90°, because the strips are unequal.
• Equal strips crossing at their middles at 90° — correct. Yes! Equal and bisecting give four right angles. The right angle between the strips makes all sides equal.
• Equal strips crossing at 90°, but not at their middles. The strips must bisect each other. Otherwise the thread does not make a rectangle at all.
• A square. A square needs the strips to cross at 90°, not 60°.
• A rectangle that is not a square — correct. Yes! Equal strips that bisect each other make a rectangle. At 60° the sides are 5 cm and 8.7 cm, so it is not a square.
• A quadrilateral that is not a rectangle. Equal diagonals that bisect each other always make a rectangle, whatever the angle.
• OA = OC, OB is common and AB = CB, so SSS — correct. Yes! The diagonals bisect each other (OA = OC), OB is shared, and the sides AB and CB of a square are equal.
• Because ∠BOA = ∠BOC. That is what we want to show. We cannot use it to prove the triangles congruent.
• Because the two triangles look alike. Looking alike is not a proof. We need matching sides: SSS.