Triangle in a circle.

Geometry Level 2

An equilateral triangle A B C ABC with A B = 2 3 AB = 2\sqrt{3} is inscribed in a circle. D D is the midpoint of A B AB . O O is the center of the circle. Find O D OD .


The answer is 1.

Shahriar Rizvi by Shahriar Rizvi , 20, Bangladesh

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2 solutions

Marta Reece
Feb 25, 2017

In a triangle A O D AOD we have ∠ D A O = 6 0 ∘ 2 = 3 0 ∘ \angle DAO=\frac{60^\circ}{2}=30^\circ , ∠ A D O = 9 0 ∘ \angle ADO=90^\circ , and A D = 1 2 × A C = 3 AD=\frac{1}{2}\times AC=\sqrt{3} , therefore:

O D = 3 × t a n ( 3 0 ∘ ) = 1 OD=\sqrt{3}\times tan(30^\circ)=1

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how do you know that the answer must be sqrt3 x tan30?

Pi Han Goh - 4 years, 3 months ago

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I have included the reasoning behind the formula. I hope this helps.

Marta Reece - 4 years, 3 months ago

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