OG详解-OG8 数学1 Q27

The correct answer is 2843​​.  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 AMO 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 AM is  2842​​ , or 142, cm. Additionally, triangle AMO has angles measuring 30°, 60°, and 90°, where the measure of angle OMA is 90°  and the measure of angle OAM is 30°. It follows that the length of side MO is half the length of hypotenuse AO, and the length of side AM is √3​  times the length of side MO. It’s given that AO = w√3​ cm. Therefore, MO =  w√32​​ cm and AM = w√3​√32​cm, which is equivalent to AM = 3w​2​  cm. Since AM = 142 cm, it follows that  3w​2​ = 142. Multiplying both sides of this equation by 2 yields 3w = 284. Dividing both sides of this equation by 3 yields w= 2843​​. Note that 284/3, 94.66, and 94.67 are examples of ways to enter a correct answer.