Question

ABC is an equilateral triangle inscribed in a circle with AB = 5 cm. Let the bisector of angle A meet BC in X and the circle in Y. What is the value of AX.AY ?

(a) 16 cm² 

(b) 20 cm² 

(c) 25 cm² 

(d) 30 cm²

Answer 1

Answer : 25 cm² 

∵ In an equilateral ∆; angle bisector AX bisects the base BC at X 

∴ BX = CX = \(\frac{5}{2}\) cm  

AX = \(\sqrt{5^2 -(5/2)^2} \) 

= \(\sqrt{25-\frac{25}{4}} \) 

= \(\sqrt{\frac{75}{4}} = \frac{5\sqrt{3}}{2}\) 

AY and BC being the chords of the circle, 

AX . XY = BX . XC 

⇒ \(\frac{5\sqrt{3}}{2}\) ⋅ XY = \(\frac{5}{2}\). \(\frac{5}{2}\) 

⇒ XY = \(\frac{5}{2\sqrt{3}}\) 

∴ AX . AY =  \(\frac{5\sqrt{3}}{2}\) . ( \(\frac{5\sqrt{3}}{2}\) + \(\frac{5}{2\sqrt{3}}\)) 

= \(\frac{75}{4}\) + \(\frac{25}{4}\) = \(\frac{100}{4}\) 

= 25 cm²