Question 40 of 50advanced🔍 AnalyzeLong Answer5 marks

Show that rectangle is the only parallelogram that can be inscribed in a circle.

Correct Answer

Let ABCD be a parallelogram inscribed in a circle. Since all its vertices lie on the circle, ABCD is a cyclic quadrilateral.

By Theorem 11, the sum of two opposite angles of a cyclic quadrilateral is 180∘180^\circ. Therefore,

∠A+∠C=180∘\angle A + \angle C = 180^\circ

and

∠B+∠D=180∘.\angle B + \angle D = 180^\circ.

In any parallelogram, opposite angles are equal. So,

∠A=∠C\angle A = \angle C

and

∠B=∠D.\angle B = \angle D.

Substituting ∠A=∠C\angle A = \angle C into the first equation, we get

∠A+∠A=180∘,\angle A + \angle A = 180^\circ,

which means

2∠A=180∘.2\angle A = 180^\circ.

Thus,

∠A=90∘.\angle A = 90^\circ.

This implies ∠C=90∘\angle C = 90^\circ. Similarly, ∠B=∠D=90∘\angle B = \angle D = 90^\circ.

Since all angles of the parallelogram are 90∘90^\circ, it must be a rectangle. Conversely, a rectangle can be inscribed in a circle as its opposite angles sum to 180∘180^\circ (by Theorem 12). Hence, a rectangle is the only parallelogram that can be inscribed in a circle.

Exercise: END-OF-CHAPTER EXERCISES | Q: *14 | (Page 24)
For More Understanding

Explanation

The proof logically combines the angle properties of a cyclic quadrilateral and a parallelogram. Theorem 11 establishes that opposite angles of a cyclic quadrilateral are supplementary (sum to 180∘180^\circ). A fundamental property of parallelograms is that their opposite angles are equal. The only way an angle can be equal to another angle while simultaneously summing to 180∘180^\circ with it is if both angles are 90∘90^\circ. Thus, all angles must be right angles, making the parallelogram a rectangle. The converse is verified using Theorem 12, confirming that rectangles can indeed be inscribed in a circle.

Solution Steps

  1. Step 1: Assume a parallelogram ABCD is inscribed in a circle, making it a cyclic quadrilateral.

  2. Step 2: Apply Theorem 11: Opposite angles of a cyclic quadrilateral sum to 180∘180^\circ (∠A+∠C=180∘\angle A + \angle C = 180^\circ).

  3. Step 3: Apply parallelogram property: Opposite angles are equal (∠A=∠C\angle A = \angle C).

  4. Step 4: Combine the equations to get 2∠A=180∘2\angle A = 180^\circ, yielding ∠A=90∘\angle A = 90^\circ and ∠C=90∘\angle C = 90^\circ. Similarly, ∠B=∠D=90∘\angle B = \angle D = 90^\circ.

  5. Step 5: Conclude that a parallelogram with all 90∘90^\circ angles is a rectangle, and confirm the converse using Theorem 12.

PreviousNext