Show that rectangle is the only parallelogram that can be inscribed in a circle.
Step-by-step solution
To find: Show that ABCD is a rectangle (and that every rectangle can be inscribed)
Idea: Two facts about the same pair of opposite angles: in a parallelogram they are equal; in a cyclic quadrilateral they add up to 180°. Both together force 90°.
- ABCD is a parallelogram, so its opposite angles are equal: ∠A = ∠C.½ mark
- ABCD is cyclic, so ∠A + ∠C = 180° (opposite angles of a cyclic quadrilateral).½ mark
- So 2∠A = 180°, ∠A = ∠C = 90°. Adjacent angles of a parallelogram are supplementary, so ∠B = ∠D = 90° too. ABCD is a rectangle.1 mark
- Conversely, in any rectangle ∠A + ∠C = 90° + 90° = 180°, so a rectangle can always be inscribed in a circle (a quadrilateral with a pair of opposite angles adding to 180° is cyclic). Hence the rectangle is the only parallelogram that can be inscribed in a circle.1 mark
Answer to write in the exam
Let parallelogram ABCD be inscribed in a circle.
∠A = ∠C (opposite angles of a parallelogram)
∠A + ∠C = 180° (opposite angles of a cyclic quadrilateral)
⇒ 2∠A = 180° ⇒ ∠A = ∠C = 90°; ∠B = 180° − ∠A = 90° = ∠D
⇒ ABCD is a rectangle.
Conversely, for a rectangle ∠A + ∠C = 180° ⇒ it is cyclic.
∴ A rectangle is the only parallelogram that can be inscribed in a circle.
Common mistakes that cost marks
- Proving only that a rectangle is cyclic. The question asks why no other parallelogram is.
- Stopping at ∠A = 90° without showing all four angles are right angles.
- Saying a square is a different case: a square is a special rectangle, so it is included.
How this can come in the exam
Assertion (A): A rhombus that is not a square cannot be inscribed in a circle.
Reason (R): A parallelogram inscribed in a circle must be a rectangle.
- Both A and R are true, and R is the correct explanation of A.
- Both A and R are true, but R is not the correct explanation of A.
- A is true, but R is false.
- A is false, but R is true.
Show answer
(A) Both A and R are true, and R is the correct explanation of A.
A rhombus is a parallelogram; if cyclic it would be a rectangle, and a rhombus that is a rectangle is a square. So R is true and explains A.
Try one yourself
Can a rhombus with one angle 60° be inscribed in a circle? Give a reason.
Show answer
No. Its opposite angles are 60° and 60°, adding to 120°, not 180°. (A rhombus is a parallelogram, and only rectangles among parallelograms are cyclic.)
More questions like this
- 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.
- Consider all chords of a circle of a fixed length. What is the shape formed by the midpoints of all these chords?
- In a circle with centre O, chords AB and AC are congruent. Explain why this statement is true: “The centre of the circle lies on the angle bisector of ∠BAC”.
- Two parallel chords of lengths 10 cm and 24 cm are on the same side of the centre of a circle. The distance between the chords is 7 cm. Find the radius of the circle.
- A regular hexagon is inscribed in a circle of radius r. Find the length of the sides of the hexagon and the distance of each side from the centre of the circle.