Circles • Topic 3 of 4

Cyclic Quadrilateral

A cyclic quadrilateral has all four vertices on one circle, and its defining property is that opposite angles are supplementary: ∠A + ∠C = 180° and ∠B + ∠D = 180°. An immediate corollary is the exterior-angle rule — an exterior angle equals the interior opposite angle. CAT loves to test whether you can spot that a quadrilateral is cyclic: if opposite angles add to 180°, or if two points subtend equal angles on the same side of a segment, the four points are concyclic. For side lengths, Ptolemy’s theorem says that in a cyclic quadrilateral the product of the diagonals equals the sum of the products of opposite sides: AC × BD = AB × CD + AD × BC. Area is given by Brahmagupta’s formula √[(s−a)(s−b)(s−c)(s−d)] where s is the semiperimeter. These two formulas convert a fearsome-looking figure into clean arithmetic.

✅ Solved examples

1. In cyclic quadrilateral ABCD, ∠A = 70° and ∠B = 95°. Find ∠C and ∠D.
∠C = 180° − 70° = 110°; ∠D = 180° − 95° = 85°.
2. An exterior angle of a cyclic quadrilateral at vertex A is 110°. Find the interior opposite angle ∠C.
Exterior angle = interior opposite angle ⇒ ∠C = 110°.
3. A cyclic quadrilateral has sides 3, 4, 5, 6. Find its area by Brahmagupta’s formula.
s = (3+4+5+6)/2 = 9. Area = √[(9−3)(9−4)(9−5)(9−6)] = √(6×5×4×3) = √360 = 6√10.
4. In a cyclic quadrilateral, AB = 5, BC = 6, CD = 5, DA = 6, and one diagonal AC = 7. Use Ptolemy to find BD.
AC × BD = AB×CD + AD×BC ⇒ 7 × BD = 5×5 + 6×6 = 25 + 36 = 61 ⇒ BD = 61/7.

✏️ Practice — try these, take hints as needed

1. In cyclic quadrilateral ABCD, ∠A = 4x and ∠C = 5x. Find x.
Opposite angles supplementary.
4x + 5x = 180.
9x = 180.
x = 20° (∠A = 80°, ∠C = 100°)
2. A cyclic quadrilateral has all sides equal to 5. Find its area (it is a square).
Equal sides on a circle ⇒ square.
Area = side².
5².
25
3. Sides of a cyclic quadrilateral are 7, 7, 7, 7? No — use 1, 2, 3, 4? Use sides 5, 5, 5, 5 with semiperimeter check Brahmagupta on 2,2,2,2.
s = sum/2.
All equal ⇒ (s−a) equal.
Area = (s−a)² when all sides equal.
4 (square of side 2, area = side² = 4)
4. In cyclic quadrilateral ABCD, ∠B = 3∠D. Find ∠D.
∠B + ∠D = 180°.
3∠D + ∠D = 180°.
4∠D = 180°.
∠D = 45°
5. A cyclic quadrilateral has diagonals 9 and 8, with AB×CD = 30. Find AD×BC using Ptolemy.
AC×BD = AB×CD + AD×BC.
9×8 = 30 + AD×BC.
72 − 30.
AD×BC = 42

📝 Topic test — 8 questions

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