GreenlightTestPrep wrote:

In the above diagram, the circle inscribes the larger equilateral, and it circumscribes the smaller equilateral triangle. If the area of the smaller triangle is √3, what is the area of the larger triangle?

A) 9π - 16√3

B) 4√3

C) 8√3

D) 16√3

E) 16π - 2√3

We're told that the area of the smaller triangle is √3

USEFUL FORMULA:

Area of an equilateral triangle = (√3)(side²)/4 So, we can write: (√3)(side²)/4 = √3

Divide both sides by √3 to get: (side²)/4 = 1

Multiply both sides by 4 to get: side² = 4

Solve:

side = 2So, each side of the smaller equilateral triangle has length

2Using this information, we can create a 30-60-90 triangle (in

blue)

We can now compare this blue 30-60-90 triangle with the

BASE 30-60-90 triangleBy the property of similar triangles, we know that the ratios of corresponding sides will be equal.

That is:

1/

√3 =

r/

2Cross multiply to get: (√3)(r) = 2

Solve: r = 2/√3

So, the RADIUS of the circle = 2/√3

We'll add this information to our diragram

At this point, we can focus our attention on the GREEN 30-60-90 triangle

Since we already know that the RADIUS of the circle = 2/√3, we can apply the property of similar triangles again.

The ratios of corresponding sides will be equal.

So, we get: (

2/√3)/

1 = (

x)/

√3 Cross multiply to get: (2/√3)(√3) = (x)(1)

Simplify: x = 2

Since x = HALF the length of one side of the larger triangle, we know that the ENTIRE length = 4

What is the area of the larger triangle? We'll re-use our formula that says: area of an equilateral triangle = (√3)(side²)/4

Area = (√3)(4²)/4

= (√3)(16)/4

= 4√3

Answer: B

Cheers

Brent

_________________

Brent Hanneson – Creator of greenlighttestprep.com

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