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The number of rectangles that can be formed by joining four vertices of a 12- sided regular polygon ABCDEFGHIJKL is:
Correct answer is '15'. Can you explain this answer?
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The number of rectangles that can be formed by joining four vertices o...
We get three kinds of rectangles.
Type 1: Rectangles with two opposite sides that are formed by consecutive vertices of the 12 sided polygon. These are same as the rectangles with two sides that are formed by vertices of the polygon that are 5 vertices apart. There are 6 such rectangles.
Type 2: Rectangles with two opposite sides that are formed by vertices of the polygon that are 2 vertices apart. These are same as the rectangles with two opposite sides that are formed by vertices of the polygon that are 4 vertices apart. There are 6 such rectangles.
Type 3: Rectangles with two sides that are formed by vertices of the polygon that are 3 vertices apart. There are 3 such rectangles. (These are squares.) Hence, 6 + 6 + 3 = 15 rectangles can be formed.
Answer: 15
Most Upvoted Answer
The number of rectangles that can be formed by joining four vertices o...
Way of doing it is choosing 2 2 vertices from both halves of the polygon which means 6C2
6C2=6!/2!*4!=15
6C2=6!/2!*4!=15
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The number of rectangles that can be formed by joining four vertices o...
Understanding the Problem
To find the number of rectangles that can be formed by joining four vertices of a 12-sided regular polygon (dodecagon), we need to recognize the properties of rectangles and the arrangement of vertices in a polygon.
Properties of Rectangles
- A rectangle is defined by two pairs of opposite sides that are equal in length and parallel.
- In a regular polygon, vertices that form a rectangle must be opposite each other.
Choosing Vertices
- In a 12-sided polygon, the vertices can be labeled as A, B, C, D, E, F, G, H, I, J, K, L.
- To form a rectangle, we need to select two pairs of opposite vertices.
Counting Opposite Pairs
- The dodecagon has 12 vertices, and opposite pairs can be formed by selecting any vertex and its directly opposite vertex.
- The opposite pairs in a 12-sided polygon are:
- (A, G), (B, H), (C, I), (D, J), (E, K), (F, L)
- There are 6 opposite pairs.
Selecting Pairs
- To form a rectangle, we need to choose 2 out of these 6 opposite pairs.
- The number of ways to choose 2 pairs from 6 pairs is calculated using combinations.
Calculating Combinations
- The formula for combinations is C(n, k) = n! / [k! * (n-k)!]
- Here, n = 6 (opposite pairs), k = 2 (pairs to choose).
- C(6, 2) = 6! / (2! * 4!) = 15.
Conclusion
Thus, the number of rectangles that can be formed by joining four vertices of a 12-sided regular polygon is 15.
To find the number of rectangles that can be formed by joining four vertices of a 12-sided regular polygon (dodecagon), we need to recognize the properties of rectangles and the arrangement of vertices in a polygon.
Properties of Rectangles
- A rectangle is defined by two pairs of opposite sides that are equal in length and parallel.
- In a regular polygon, vertices that form a rectangle must be opposite each other.
Choosing Vertices
- In a 12-sided polygon, the vertices can be labeled as A, B, C, D, E, F, G, H, I, J, K, L.
- To form a rectangle, we need to select two pairs of opposite vertices.
Counting Opposite Pairs
- The dodecagon has 12 vertices, and opposite pairs can be formed by selecting any vertex and its directly opposite vertex.
- The opposite pairs in a 12-sided polygon are:
- (A, G), (B, H), (C, I), (D, J), (E, K), (F, L)
- There are 6 opposite pairs.
Selecting Pairs
- To form a rectangle, we need to choose 2 out of these 6 opposite pairs.
- The number of ways to choose 2 pairs from 6 pairs is calculated using combinations.
Calculating Combinations
- The formula for combinations is C(n, k) = n! / [k! * (n-k)!]
- Here, n = 6 (opposite pairs), k = 2 (pairs to choose).
- C(6, 2) = 6! / (2! * 4!) = 15.
Conclusion
Thus, the number of rectangles that can be formed by joining four vertices of a 12-sided regular polygon is 15.
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The number of rectangles that can be formed by joining four vertices of a 12- sided regular polygon ABCDEFGHIJKL is:Correct answer is '15'. Can you explain this answer?
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