What the area of the equilateral triangle inscribed in circle C? 1) A

Question

What the area of the equilateral triangle inscribed in circle C? 1) Area of the smallest equilateral triangle enclosing the circle C is 9√3 2) Radius of circle C is √3 Source: www.GMATinsight.com

Archit3110 · Answer

target find area of equilateral ∆ inscribed in circle C
#1
Area of the smallest equilateral triangle enclosing the circle C is  9√3 
area of equilateral ∆ = √3/4 * side^2
9√3= √3/4 * side ^2
side = 6
height of equilateral ∆ = V3/2 * side ; √3/2* 6 ; 3√3

radius of an incircle in an equilateral ∆ ; 1/3 * height of equilateral ∆ ; 1/3 *3V3 ; √3
now side of equilateral ∆ inside circle of radius √3 can be determined
equilateral ∆ side = √3* radius of circle ; √3*√3 ; 3
area of equilateral ∆ = V3*3*3/4 ; 9√3/4 ; sufficient

#2
Radius of circle C is √3
equilateral ∆ side = √3* radius of circle ; √3*√3 ; 3
area of equilateral ∆ = √3*3*3/4 ; 9√3/4 ; sufficient

option D

swim2109 · Answer

I could not understand the steps highlightd in red? Can you explain a bit more or paste a picture?

SiddharthR · Answer

Hello  swim2109

I got this question wrong too but figured out how to solve it. Maybe I can help out here

I am going to quote two formulas I know from the GMAT Club Math Book which helped me solve this question

"1) Area of a triangle with sides a,b,c with an inscribed circle of radius r = Pr/2; P is the perimeter of the triangle (a+b+c) and r is the radius of the circle

2) Area of a triangle with sides a,b,c circumscribed by a circle with radius R = abc/4R (a,b,c sides of a triangle and R is the radius of the circle
"

Given: Equilateral triangle circumscribed by a circle. Need to find area of this triangle

If length of this triangle is say 'x' then area of the equilateral triangle is  * (x^2). I started with Statement II (this also helps with understanding Statement I easily)

Statement II) From the quote "2" above, we know that the area of a triangle circumscribed by a circle is given by abc/4R. Since we are dealing with equilateral triangles, this is a^3/4R (calling sides of the triangle as 'a') ---> (a)

We know that area of equilateral triangle with side a is given by  * (a^2) --> (b)

Equating (a) & (b), we have  * (a^2) = a^3/4R --> (c) (R = sqrt(3) from Statement II)

Solving (c) should give us the side of the triangle 'a' that helps us to find the area of the triangle. Hence Statement II is sufficient

Statement I) Again assuming side of triangle is 'a'

* (a^2) = 9sqrt(3) --> (d)

Solve for 'a' (Side of Equilateral Triangle)

Now substitute this in the quoted formula '1': Area of Triangle = Pr/2
9sqrt(3) ("From Statement II") =  /2 --> This should give us the radius of the circle. Then following the same method described for statement 1 you should be able to find the area of the equilateral triangle that is circumscribed by a circle of radius 'r'

What the area of the equilateral triangle inscribed in circle C? 1) A

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What the area of the equilateral triangle inscribed in circle C? 1) A

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