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Geometry and Trigonometry Difficulty: Hard

The perimeter of an equilateral triangle is $852$ centimeters. The three vertices of the triangle lie on a circle. The radius of the circle is $w StartRoot 3 EndRoot$ centimeters. What is the value of $w$ ?

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Explanation

The correct answer is $StartFraction 284 Over 3 EndFraction$ . Since the perimeter of a triangle is the sum of the lengths of its sides, and the given triangle is equilateral, the length of each side is $StartFraction 852 Over 3 EndFraction$ , or $284$ , centimeters (cm). Right triangle $A M O$ can be formed, where $M$ is the midpoint of one of the triangle’s sides, $A$ is one of this side’s endpoints, and $O$ is the center of the circle. It follows that $A M$ is $StartFraction 284 Over 2 EndFraction$ , or $142$ , cm. Additionally, triangle $A M O$ has angles measuring $30 degree$ , $60 degree$ , and $90 degree$ , where the measure of angle $O M A$ is $90 degree$ and the measure of angle $O A M$ is $30 degree$ . It follows that the length of side $M O$ is half the length of hypotenuse $A O$ , and the length of side $A M$ is $StartRoot 3 EndRoot$ times the length of side $M O$ . It’s given that $upper A upper O equals w StartRoot 3 EndRoot$ cm. Therefore, $M O = \frac{w \sqrt{3}}{2}$ cm and $A M = \frac{w \sqrt{3} \sqrt{3}}{2}$ cm, which is equivalent to $A M = \frac{3 w}{2}$ cm. Since $upper A upper M equals 142$ cm, it follows that $\frac{3 w}{2} = 142$ . Multiplying both sides of this equation by $2$ yields $3 w equals 284$ . Dividing both sides of this equation by $3$ yields $w = \frac{284}{3}$ . Note that 284/3, 94.66, and 94.67 are examples of ways to enter a correct answer.