The perimeter of an equilateral triangle is 852 centimeters. The three vertices of the triangle lie on a circle. The

Question

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 3 centimeters. What is the value of w ?

Accepted Answer

The correct answer is 284 3 . 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 852 3 , or 284 , centimeters (cm). Right triangle A M O can be formed, where M is the midpoint of one of the triangle&rsquo;s sides, A is one of this side&rsquo;s endpoints, and O is the center of the circle. It follows that A M is 284 2 , or 142 , cm. Additionally, triangle A M O has angles measuring 30 &deg; , 60 &deg; , and 90 &deg; , where the measure of angle O M A is 90 &deg; and the measure of angle O A M is 30 &deg; . 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 3 times the length of side M O . It&rsquo;s given that A O = w 3 cm. Therefore, M O = w 3 2 cm and A M = w 3 3 2 cm, which is equivalent to A M = 3 w 2 cm. Since A M = 142 cm, it follows that 3 w 2 = 142 . Multiplying both sides of this equation by 2 yields 3 w = 284 . Dividing both sides of this equation by 3 yields w = 284 3 . Note that 284/3, 94.66, and 94.67 are examples of ways to enter a correct answer.

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