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# Please Solve This Math "Thinking" Problem?

Topic: Area of triangles problem solving
July 20, 2019 / By Dacey
Question: I'm making a "thinking" math problem for my students ("thinking" because it's kind of a tricky question), and I got one from my friend but I don't know how to solve it! XD So if you could please solve it for me and tell me how you got the answer, that would be greatly appreciated. Problem: If a triangle has a perimeter of 3, what is the largest possible area?

August | 4 days ago
First of all, we know that the answer must always be less that 1 square unit. We know this because the area of a triangle is base times height. Given a base of 1 unit, the remaining perimeter measure can only total 2 units. Dividing that in half to gain the greatest possible height gives us sides of 1 unit each, which when brought together at a common vertex give us a height of less that 1 unit. In order to achieve a height of 1 unit and still maintain a perimeter of 3 units, we would have to shorten the base to less than 1 unit. Therefore, our answer will always be less that 1 square unit. The maximum area given a perimeter of 3, or for any given perimeter, is achieved by forming an equilateral triangle. In this case an area of .866 square units is formed.
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First of all, we know that the answer must always be less that 1 square unit. We know this because the area of a triangle is base times height. Given a base of 1 unit, the remaining perimeter measure can only total 2 units. Dividing that in half to gain the greatest possible height gives us sides of 1 unit each, which when brought together at a common vertex give us a height of less that 1 unit. In order to achieve a height of 1 unit and still maintain a perimeter of 3 units, we would have to shorten the base to less than 1 unit. Therefore, our answer will always be less that 1 square unit. The maximum area given a perimeter of 3, or for any given perimeter, is achieved by forming an equilateral triangle. In this case an area of .866 square units is formed.