Theory question 104
Circle TheoremsIn the diagram, ABCD is a circle O. The quadrilateral OBCD is a rhombus such that ∠ADO = ∠OBA = y and ∠BAD = t. Find (a) the value of t; (b) the value of y; (c) ∠ADC.
Show the model answer
Model answer
OB = OC = OD (radii) and OBCD is a rhombus, so OB = BC = CD = DO = OC: triangles OBC and OCD are equilateral
∠BOD (through C) = 60° + 60° = 120°
(a) t = ∠BAD = ½ × 120° = 60° (angle at centre is twice angle at circumference)
(b) OA = OB = OD, so ∠OAB = ∠OBA = y and ∠OAD = ∠ODA = y; t = 2y ⇒ y = 30°
(c) ∠ADC = ∠ADO + ∠ODC = 30° + 60° = 90°
Explanation
Every side of the rhombus equals the radius, so look for equilateral triangles.