Theory question 108
Circle Theorems(a) In the diagram, A, B, C and D are points on the circle with centre O. ∠DAB = 64° and |AD| = |DC|. Find the size of the angle marked x.
(b) Two consecutive even numbers are such that one-third of the larger exceeds one-fourth of the smaller by 7. Find the numbers.
Show the model answer
Model answer
(a) ∠DCB = 180° − 64° = 116° (cyclic quadrilateral); ∠ACB = 90° (AB is a diameter) ⇒ ∠ACD = 26°
AD = DC ⇒ ∠DAC = ∠DCA = 26°; x = ∠ADC = 180° − 52° = 128°
(b) Numbers n and n + 2: (n + 2)/3 − n/4 = 7 ⇒ 4n + 8 − 3n = 84 ⇒ n = 76
The numbers are 76 and 78
Explanation
AB passes through O, so the angle in the semicircle is 90°.