Multiple choice

The radii ${ r }{ 1 },{ r }{ 2 },{ r }_{ 3 }$ escribed circle of the traingle $ABC$ are in H.P. If its area is $24$ square cm and its perimeter is $24$ cm, then lengths of its sides are

  1. $4,6,8$
  2. $3,9,11$
  3. $6,8,10$

  4. None of these
Reveal answer Fill a bubble to check yourself
C Correct answer
Explanation

Area = 24, perimeter = 24, so semi-perimeter s = 12. r = Area/s = 24/12 = 2. The exradii are r1 = Area/(s-a), r2 = Area/(s-b), r3 = Area/(s-c). Since they are in HP, (s-a), (s-b), (s-c) are in AP. This implies a, b, c are in AP. For sides 6, 8, 10, s = 12, area = sqrt(12*6*4*2) = 24. This matches.

AI explanation

The area of a triangle is the product of its semi-perimeter (s) and the radius of its incircle (r). Given an area of 24 square cm and a perimeter of 24 cm (so s = 12 cm), the inradius is 24 / 12 = 2 cm. The exradii (r1, r2, r3) are related to the area by the formulas Area / (s - a), Area / (s - b), and Area / (s - c), which are the reciprocals used to test if they are in a harmonic progression. For the sides 6, 8, and 10, the exradii are 24 / (12 - 6) = 4, 24 / (12 - 8) = 6, and 24 / (12 - 10) = 12; since the reciprocals 1/4, 1/6, and 1/12 are in an arithmetic progression, the exradii are in a harmonic progression.

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