If area of a circle inscribed in an equilateral triangle is 48π square units, then perimeter of the triangle is
(a)17√3 units
(b)36 units
(c)72 units
(d)48√3 units
Answers
Answer:
The Perimeter of equilateral triangle is 72 units.
Among the given options option (c) 72 units is the correct answer.
Step-by-step explanation:
Given :
Area of circle ,A = 48π sq.units
πr² = 48π
r² = 48
r = √48
r = √16 × 3
r = 4√3
Radius of a circle = 4√3 units
Let ABC is an equilateral triangle of side ‘a’ cm.
Join OA, OB, and OC. O is the incentre of a circle.
OP, OR & OQ are Radius of a circle and they are equal .
Let OP = OR = OQ = r & AB = BC = AC = a
Area of ∆AOB + Area of ∆BOC + Area of ∆AOC = Area of ∆ABC
(½ × AB × OR) + (½ × BC × OP) + (½ × AC × OQ) = √3/4 × a²
[Area of ∆ = ½ × base × height , Area of equilateral ∆ = √3/4 side²]
√3/4× a² = (½ × a × r) + (½ × a × r) + (½ × a × r)
√3/4× a² = (½ × a × r)(1 + 1 +1 )
√3/4× a² = (½ × a × r) × 3
√3/4× a² = (½ × a × 4√3) × 3
√3/4× a² = 6a/√3
a²/a = 6/√3 × 4/√3
a = 6 × 4 = 24
a = 24
Side of equilateral triangle = 24 units
Perimeter of equilateral triangle = 3 × side
Perimeter of equilateral triangle = 3 × 24 = 72 units
Hence, the Perimeter of equilateral triangle is 72 units.
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Answer :
(c) 72 Units
Explanation :
Kindly check out the attachment for Figure
Area of the circle is given as 48π.
⇒ πr² = 48π
⇒ r² = 48
⇒ r = 4√3
Now, it is clear that ON⊥BC. So, ON is the height of ΔOBC corresponding to BC.
Area of ΔABC = Area of ΔOBC + Area of ΔOCA + Area of ΔOAB = 3 × Area of ΔOBC
√3/4 * a² = 3 * ½ * BC * ON
√3/4 * a² = 3 * ½ * a * r
√3/4 * a² = 3 * ½ * a * 4√3
a = 24 unit²
Thus, perimeter of the equilateral triangle = 3 × 24 units = 72 units
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