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FRESH GMAT CLUB TESTS QUESTION
The length of each side of equilateral triangle M is 3 times the length of each side of equilateral triangle N. What is the perimeter of equilateral triangle N ?
(1) The ratio of the area of equilateral triangle M to the area of equilateral triangle N is 9:1
(2) The sum of the circumferences of circles circumscribing triangles M and N is \(12\pi\)
1) Ratio of length of sides= 3:1, therefore ratio of areas is 3^2:1^2 or 9:1. Repetition of info given in question.
Insufficient.2) Circumference of a circle is \(2\pi r\),
As the highest power of r is one and \(2\pi\) is constant which will cancel out in a fraction, the ratio of circumference of triangles is equal to ratio of the circumscribed radii of the equilateral triangles, the radii are proportional to the sides of equilateral triangles (explanation given below). Hence, circumferences of circles circumscribing triangles N= \(12\pi\) * \(\frac{1}{4}\)= \(3\pi\). From this we can easily calculate the side of equilateral triangle and 3 times side is perimeter of equilateral triangle N.
Sufficient.Answer is B.For problem solving questions, the relationship between side of an equilateral triangle and radius of circumscribed circle is given below:
For an equilateral triangle with side S, the median is \(\frac{\sqrt{3}S}{2}\) and \(\frac{1}{3}\) of median is radius of circumscribed circle. Therefore, radius of circumscribed circle is \(\frac{S}{\sqrt{3}}\)Please hit +1 Kudos if you liked the answer.