Multiple choice

The distance of the incentre of the triangle $ABC$ from $A$ is $4R\sin{\left(\dfrac{A}{2}\right)}$

  1. True

  2. False
Reveal answer Fill a bubble to check yourself
B Correct answer
Explanation

The distance of the incentre from vertex A is r / sin(A/2). Since r = 4R sin(A/2) sin(B/2) sin(C/2), the expression 4R sin(A/2) is incorrect.

AI explanation

If I is the incentre, in triangle AIC the angle AIC equals 90 degrees plus half of angle B, and the length IC equals the radius r divided by the sine of half angle C. Using the triangle angle sum identity and substituting r equals 4R times the sine of half A times the sine of half B times the sine of half C, the distance AI simplifies exactly to r divided by the sine of half angle A. This expands to 4R times the sine of half angle B times the sine of half angle C, not 4R times the sine of half angle A, making the statement false.

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