Multiple choice

The distance of the circumcentre of the acute angled $\angle ABC$ from the sides $BC,CA$ and $AB$ are in the ratio ?

  1. $a\sin A: b \sin B: c \sin C$
  2. $\cos A$$: \cos B$$: \cos C$
  3. $a\cot A: b \cot B: c \cot C$
  4. $none\ of\ these$
Reveal answer Fill a bubble to check yourself
B Correct answer
Explanation

The distances from the circumcenter to the sides of a triangle are given by R cos A, R cos B, and R cos C, where R is the circumradius. Therefore, the ratio of these distances is cos A : cos B : cos C.

AI explanation

Let O be the circumcentre of acute triangle ABC, and let R be its circumradius. The distance from O to side BC is the length of the segment from the center to the chord, given by R cos A. Similarly, the distances from O to sides CA and AB are R cos B and R cos C. Removing the common scalar R, the ratio of the distances is cos A : cos B : cos C.

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