Math Problem Statement
Begin by analyzing the problem: If 100 regular triangles with side lengths of 1 unit are lined up in a row, sharing one side with the adjacent triangle, what will the perimeter of the shape be? Consider the following questions: What will the perimeter of the shape be? What shape do the 100 triangles create? Can you create a rule for finding the perimeter of a shape created by any number of triangles?can you show me how to solve with pictures
Solution
I see the problem involves determining the perimeter of a shape formed by lining up 100 equilateral triangles, where each triangle shares one side with the next. Let's walk through the solution step by step.
Analysis:
Each regular triangle has three equal sides. When you line up triangles in a row, the side they share with adjacent triangles no longer contributes to the perimeter. The key idea is to count how many sides are still exposed.
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For 1 triangle: The perimeter is sides.
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For 2 triangles: The shared side between the two triangles removes 1 side from the perimeter, leaving:
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For 3 triangles: Two sides are shared between adjacent triangles, so:
General Rule:
If you have triangles lined up, the perimeter can be calculated as: This rule works because there are always 2 sides for each triangle plus the unshared sides at the ends.
Solving for 100 triangles:
Using the formula: So the perimeter for 100 triangles is 201 units.
Would you like a visual representation or need further clarification? Let me know if you want to explore any of the following:
- How is the perimeter of other polygonal shapes calculated when combined in sequences?
- What happens when the triangles are not equilateral or vary in size?
- Can the rule be adjusted for different arrangements, like stacking triangles in rows?
- What mathematical reasoning underlies the generalization to triangles?
- What if you try to enclose the shape with other polygons?
Tip:
In any geometric arrangement, identify the shared sides or areas, as this reduces the overall perimeter or surface area.
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Math Problem Analysis
Mathematical Concepts
Geometry
Perimeter Calculation
Sequences
Formulas
Perimeter = 2n + 1
Theorems
Generalization of perimeter formula for linear arrangements of regular polygons
Suitable Grade Level
Grades 6-8
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