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Find the number of non-congruent rectangles that can be found on a chessboard normal 8 x 8 chessboard,
- a)24
- b)36
- c)48
- d)None of these
Correct answer is option 'B'. Can you explain this answer?
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Find the number of non-congruent rectangles that can be found on a che...
Find 1 x 2 rectangles, 2 x 3 rectangles, 3 x 4 rectangles, 4 x 5 rectangles, 5 x 6 rectangles, 6 x 7 rectangles, 7 x 8 rectangles.
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Find the number of non-congruent rectangles that can be found on a che...
Explanation:
To find the number of non-congruent rectangles on an 8x8 chessboard, we can break down the problem into smaller cases and then add up the results.
Case 1: Rectangles with sides parallel to the edges of the chessboard
In this case, we can choose any two rows and any two columns to form a rectangle. There are 8 rows and 8 columns on the chessboard, so the number of rectangles in this case is 8 choose 2 for both rows and columns, which is equal to (8C2)^2 = 28^2 = 784.
Case 2: Rectangles with sides at a 45-degree angle to the edges of the chessboard
In this case, we need to consider rectangles that are formed by choosing two diagonally opposite corners of the chessboard. There are (8-1)^2 = 49 such rectangles.
Case 3: Rectangles with sides diagonal to the edges of the chessboard
In this case, we need to consider rectangles that are formed by choosing two corners that are not on the same row, column, or diagonal. There are 7 choose 2 = 21 ways to choose the two rows and the same for the columns, so the number of rectangles in this case is (7C2)^2 = 21^2 = 441.
Total number of rectangles
To find the total number of rectangles, we add up the number of rectangles in each case:
Total = Case 1 + Case 2 + Case 3
= 784 + 49 + 441
= 1274
Therefore, the number of non-congruent rectangles that can be found on an 8x8 chessboard is 1274. None of the given options (a, b, or c) match the correct answer, so the correct answer is option 'd', None of these.
To find the number of non-congruent rectangles on an 8x8 chessboard, we can break down the problem into smaller cases and then add up the results.
Case 1: Rectangles with sides parallel to the edges of the chessboard
In this case, we can choose any two rows and any two columns to form a rectangle. There are 8 rows and 8 columns on the chessboard, so the number of rectangles in this case is 8 choose 2 for both rows and columns, which is equal to (8C2)^2 = 28^2 = 784.
Case 2: Rectangles with sides at a 45-degree angle to the edges of the chessboard
In this case, we need to consider rectangles that are formed by choosing two diagonally opposite corners of the chessboard. There are (8-1)^2 = 49 such rectangles.
Case 3: Rectangles with sides diagonal to the edges of the chessboard
In this case, we need to consider rectangles that are formed by choosing two corners that are not on the same row, column, or diagonal. There are 7 choose 2 = 21 ways to choose the two rows and the same for the columns, so the number of rectangles in this case is (7C2)^2 = 21^2 = 441.
Total number of rectangles
To find the total number of rectangles, we add up the number of rectangles in each case:
Total = Case 1 + Case 2 + Case 3
= 784 + 49 + 441
= 1274
Therefore, the number of non-congruent rectangles that can be found on an 8x8 chessboard is 1274. None of the given options (a, b, or c) match the correct answer, so the correct answer is option 'd', None of these.
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Find the number of non-congruent rectangles that can be found on a che...
Yes 36 is the correct answer
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Find the number of non-congruent rectangles that can be found on a chessboard normal 8 x 8 chessboard,a)24b)36c)48d)None of theseCorrect answer is option 'B'. Can you explain this answer?
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