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An insect lives on the surface of a regular tetrahedron with edges of length 1. It wishes to travel on the surface of the tetrahedron from the midpoint of one edge to the midpoint of the opposite edge. What is the length of the shortest such trip? (Note: two edges of a tetrahedron are opposite if they have no common endpoint.)?
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An insect lives on the surface of a regular tetrahedron with edges of ...
Understanding the Tetrahedron Structure
The regular tetrahedron consists of four triangular faces, six edges, and four vertices. Each edge has a length of 1.
Identifying the Midpoints
- Let the vertices of the tetrahedron be A, B, C, and D.
- Consider the edge AB and its midpoint M1.
- The opposite edge is CD, and its midpoint is M2.
Unfolding the Tetrahedron
To find the shortest path, we can visualize unfolding the tetrahedron into a 2D plane:
1. Choose a Base: Select triangle ABC as the base.
2. Place the Midpoints: M1 (midpoint of AB) lies on edge AB, while M2 (midpoint of CD) can be extended from point C to point D.
Calculating the Shortest Path
- When the tetrahedron is unfolded, the path between M1 and M2 can be represented as a straight line across the unfolded surface.
- M1 is located at (0.5, 0) in the 2D plane.
- M2 lies at (0.5, sqrt(3)/2) after unfolding.
Distance Calculation
- The distance between points M1 and M2 is calculated using the distance formula.
- The distance = sqrt((0.5 - 0.5)² + (0 - sqrt(3)/2)²) = sqrt(0 + (sqrt(3)/2)²) = sqrt(3)/2.
Conclusion
The shortest trip for the insect from the midpoint of one edge to the midpoint of the opposite edge on the surface of a regular tetrahedron is sqrt(3)/2. This process of unfolding the shape simplifies the complex 3D path into a straightforward 2D distance calculation.
The regular tetrahedron consists of four triangular faces, six edges, and four vertices. Each edge has a length of 1.
Identifying the Midpoints
- Let the vertices of the tetrahedron be A, B, C, and D.
- Consider the edge AB and its midpoint M1.
- The opposite edge is CD, and its midpoint is M2.
Unfolding the Tetrahedron
To find the shortest path, we can visualize unfolding the tetrahedron into a 2D plane:
1. Choose a Base: Select triangle ABC as the base.
2. Place the Midpoints: M1 (midpoint of AB) lies on edge AB, while M2 (midpoint of CD) can be extended from point C to point D.
Calculating the Shortest Path
- When the tetrahedron is unfolded, the path between M1 and M2 can be represented as a straight line across the unfolded surface.
- M1 is located at (0.5, 0) in the 2D plane.
- M2 lies at (0.5, sqrt(3)/2) after unfolding.
Distance Calculation
- The distance between points M1 and M2 is calculated using the distance formula.
- The distance = sqrt((0.5 - 0.5)² + (0 - sqrt(3)/2)²) = sqrt(0 + (sqrt(3)/2)²) = sqrt(3)/2.
Conclusion
The shortest trip for the insect from the midpoint of one edge to the midpoint of the opposite edge on the surface of a regular tetrahedron is sqrt(3)/2. This process of unfolding the shape simplifies the complex 3D path into a straightforward 2D distance calculation.
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An insect lives on the surface of a regular tetrahedron with edges of length 1. It wishes to travel on the surface of the tetrahedron from the midpoint of one edge to the midpoint of the opposite edge. What is the length of the shortest such trip? (Note: two edges of a tetrahedron are opposite if they have no common endpoint.)?
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An insect lives on the surface of a regular tetrahedron with edges of length 1. It wishes to travel on the surface of the tetrahedron from the midpoint of one edge to the midpoint of the opposite edge. What is the length of the shortest such trip? (Note: two edges of a tetrahedron are opposite if they have no common endpoint.)? for JEE 2024 is part of JEE preparation. The Question and answers have been prepared according to the JEE exam syllabus. Information about An insect lives on the surface of a regular tetrahedron with edges of length 1. It wishes to travel on the surface of the tetrahedron from the midpoint of one edge to the midpoint of the opposite edge. What is the length of the shortest such trip? (Note: two edges of a tetrahedron are opposite if they have no common endpoint.)? covers all topics & solutions for JEE 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for An insect lives on the surface of a regular tetrahedron with edges of length 1. It wishes to travel on the surface of the tetrahedron from the midpoint of one edge to the midpoint of the opposite edge. What is the length of the shortest such trip? (Note: two edges of a tetrahedron are opposite if they have no common endpoint.)?.
An insect lives on the surface of a regular tetrahedron with edges of length 1. It wishes to travel on the surface of the tetrahedron from the midpoint of one edge to the midpoint of the opposite edge. What is the length of the shortest such trip? (Note: two edges of a tetrahedron are opposite if they have no common endpoint.)? for JEE 2024 is part of JEE preparation. The Question and answers have been prepared according to the JEE exam syllabus. Information about An insect lives on the surface of a regular tetrahedron with edges of length 1. It wishes to travel on the surface of the tetrahedron from the midpoint of one edge to the midpoint of the opposite edge. What is the length of the shortest such trip? (Note: two edges of a tetrahedron are opposite if they have no common endpoint.)? covers all topics & solutions for JEE 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for An insect lives on the surface of a regular tetrahedron with edges of length 1. It wishes to travel on the surface of the tetrahedron from the midpoint of one edge to the midpoint of the opposite edge. What is the length of the shortest such trip? (Note: two edges of a tetrahedron are opposite if they have no common endpoint.)?.
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