Shortcut Trick
In a cyclic quadrilateral, the sum of opposite angles is 180°.
∠A + ∠C = 180° ⇒ 3x + 9y = 180 ⇒ x + 3y = 60
∠B + ∠D = 180° ⇒ 5x + 2y = 180
Multiply first equation by 5: 5x + 15y = 300.
Subtracting second from first: (5x + 15y) − (5x + 2y) = 300 − 180 ⇒ 13y = 120.
Ratio y/x = y ÷ (60 − 3y) = (120/13) ÷ (780/13 − 360/13) = 120 ÷ 420 = 2/7.
∴ The correct answer is 2/7.
Alternate Method
Given: ∠A = 3x°, ∠B = 5x°, ∠C = 9y°, ∠D = 2y°.
Formula: Sum of opposite angles in a cyclic quadrilateral is 180°.
Since ABCD is a cyclic quadrilateral, opposite angles sum to 180°:
⇒ ∠A + ∠C = 180°
⇒ 3x + 9y = 180
⇒ x + 3y = 60 ⇒ x = 60 − 3y — (Eq. 1)
⇒ ∠B + ∠D = 180°
⇒ 5x + 2y = 180 — (Eq. 2)
Substituting (Eq. 1) into (Eq. 2):
⇒ 5(60 − 3y) + 2y = 180
⇒ 300 − 15y + 2y = 180
⇒ 300 − 13y = 180
⇒ 13y = 120 ⇒ y = 120/13
Now, finding x from (Eq. 1):
⇒ x = 60 − 3(120/13)
⇒ x = (780 − 360) / 13 = 420/13
Calculating the ratio y/x:
⇒ y/x = (120/13) ÷ (420/13) = 120 / 420
⇒ y/x = 12/42 = 2/7
∴ The correct answer is 2/7.
Additional Information
Cyclic Quadrilateral Properties
A quadrilateral whose vertices all lie on a single circle. The sum of opposite angles is always 180°.
Exterior Angle Theorem
The exterior angle formed by extending a side of a cyclic quadrilateral is equal to the interior opposite angle.
Ptolemy's Theorem
For a cyclic quadrilateral with sides a, b, c, d and diagonals p, q: p × q = ac + bd.