The radius of the incircle of an equilateral triangle is 9√3. What will be the perimeter of this equilateral triangle?
Options:
1. 54 cm
2. 162 cm
3. 108 cm
4. 81 cm
Answer (2)
The perimeter of the equilateral triangle, given the incircle radius of 9√3 cm, is 162 cm. This is calculated by first finding the side length using the relationship between the incircle radius and side length. Therefore, the correct option is 2: 162 cm.
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To solve this problem, we need to determine the perimeter of an equilateral triangle given that the radius of its incircle is 9 3 .
Understanding the Properties:
In an equilateral triangle, the radius r of the incircle is related to the side a of the triangle by the formula: r = 6 a 3 .
Substituting the Given Radius:
We are given that r = 9 3 .
Substituting this into the formula, we have: 9 3 = 6 a 3 .
Simplifying the Equation:
To isolate a , multiply both sides by 6: 6 × 9 3 = a 3 . 54 3 = a 3 .
Divide both sides by 3 : 54 = a .
Calculating the Perimeter:
The perimeter P of an equilateral triangle is given by: P = 3 a .
Substitute a = 54 into this formula: P = 3 × 54 = 162.
Therefore, the perimeter of the equilateral triangle is 162 cm . The correct answer is option 2: 162 cm.
