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Cyclic quadrilaterals · 3 marks

Show that if a rectangle is inscribed in a circle, then the point of intersection of its diagonals must lie at the centre of the circle.

Answer: ∠ABC = 90°, so the arc AC not containing B subtends 2 × 90° = 180° at the centre O: A, O, C are collinear and AC is a diameter. Likewise BD is a diameter. Two diameters meet only at the centre, so the diagonals meet at O.

Step-by-step solution

Given: Rectangle ABCD inscribed in a circle with centre O
To find: Show that AC and BD meet at O

Idea: Use the converse view of the angle in a semicircle: a chord that subtends 90° at a point of the circle subtends 180° at the centre, so it is a diameter.

ABCDO
  1. ∠ABC = 90° (angle of a rectangle). B is on the circle outside the arc AC that does not contain B, so that arc subtends ∠AOC = 2 × 90° = 180° at O.1 mark
  2. A straight angle at O means A, O, C are collinear: AC is a diameter and passes through O.½ mark
  3. In the same way ∠BAD = 90° gives ∠BOD = 180°, so BD is a diameter and passes through O.½ mark
  4. The two diagonals are different lines, so they meet at exactly one point. Both pass through O, so their point of intersection is the centre O.1 mark
Each diagonal of the inscribed rectangle subtends 90° at a vertex, so it subtends 180° at the centre and is a diameter. Both diagonals pass through the centre, so they intersect at the centre.

Answer to write in the exam

∠ABC = 90° (angle of a rectangle)

∠AOC = 2∠ABC = 180° (angle at the centre is double the angle at the circle) ⇒ A, O, C collinear ⇒ AC is a diameter

Similarly ∠BAD = 90° ⇒ ∠BOD = 180° ⇒ BD is a diameter

AC and BD both pass through O and meet in only one point

∴ The diagonals intersect at the centre O.

Common mistakes that cost marks

  • Saying “the diagonals bisect each other, so they meet at the centre” without showing that the midpoint is the centre. The key fact is that each diagonal is a diameter.
  • Using ∠AOC = 90° instead of 180°: the angle at the centre is double.
  • Proving only one diagonal is a diameter; two lines through O are needed to fix the point.

How this can come in the exam

MCQ (1 mark)

A rectangle of sides 9 cm and 12 cm is inscribed in a circle. The radius of the circle is

  1. 6 cm
  2. 7.5 cm
  3. 10.5 cm
  4. 15 cm
Show answer

(B) 7.5 cm
The diagonal is a diameter: √(81 + 144) = 15 cm, so the radius is 7.5 cm.

Try one yourself

A rectangle with sides 7 cm and 24 cm is inscribed in a circle. Find the radius and say where the centre is.

Show answer

Diagonal = √(49 + 576) = 25 cm, a diameter. Radius = 12.5 cm; the centre is the point where the diagonals cross.

More questions like this

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