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There are 4 horizontal and 4 vertical lines, parallel and equidistant to one another on a board. What is the maximum number of rectangles and squares that can be formed?
- a)16
- b)24
- c)36
- d)42
Correct answer is option 'C'. Can you explain this answer?
Verified Answer
There are 4 horizontal and 4 vertical lines, parallel and equidistant ...
4 vertical and 4 horizontal lines
Squares are: a, b, c, d, e, f, g, h, i, (abde), (bcef), (degh), (efhi), ABCD
Rectangles are: (ab), (bc), (de), (ef), (gh), (hi), (ad), (dg), (be), (eh), (cf), (fi),
(abc), (def), (ghi), (adg), (beh), (cfi),
(abcdef), (defghi), (adgbeh), (behcfi)
Required number = 14 + 22 = 36
Squares are: a, b, c, d, e, f, g, h, i, (abde), (bcef), (degh), (efhi), ABCD
Rectangles are: (ab), (bc), (de), (ef), (gh), (hi), (ad), (dg), (be), (eh), (cf), (fi),
(abc), (def), (ghi), (adg), (beh), (cfi),
(abcdef), (defghi), (adgbeh), (behcfi)
Required number = 14 + 22 = 36
Most Upvoted Answer
There are 4 horizontal and 4 vertical lines, parallel and equidistant ...
Maximum number of rectangles and squares that can be formed on a board with 4 horizontal and 4 vertical lines can be calculated as follows:
Rectangles:
- We can choose any two horizontal lines out of 4 in 4C2 ways i.e. 6 ways.
- Similarly, we can choose any two vertical lines out of 4 in 4C2 ways i.e. 6 ways.
- The total number of rectangles formed by these two horizontal and two vertical lines is 6 x 6 = 36.
Squares:
- We can choose any two horizontal lines out of 4 in 4C2 ways i.e. 6 ways.
- Similarly, we can choose any two vertical lines out of 4 in 4C2 ways i.e. 6 ways.
- Out of these 6 x 6 = 36 rectangles, we need to count only those which are squares.
- A square is formed when the distance between two horizontal lines is equal to the distance between two vertical lines.
- Since there are 4 horizontal and 4 vertical lines, there are 3 distances for each i.e. 3 horizontal distances and 3 vertical distances.
- Out of these 3 horizontal distances, only one distance is equal to each of the 3 vertical distances.
- Therefore, there are only 3 squares that can be formed by these 4 horizontal and 4 vertical lines.
- Hence, the total number of squares formed is 3.
Therefore, the maximum number of rectangles and squares that can be formed is 36 + 3 = 39, which is option C.
Rectangles:
- We can choose any two horizontal lines out of 4 in 4C2 ways i.e. 6 ways.
- Similarly, we can choose any two vertical lines out of 4 in 4C2 ways i.e. 6 ways.
- The total number of rectangles formed by these two horizontal and two vertical lines is 6 x 6 = 36.
Squares:
- We can choose any two horizontal lines out of 4 in 4C2 ways i.e. 6 ways.
- Similarly, we can choose any two vertical lines out of 4 in 4C2 ways i.e. 6 ways.
- Out of these 6 x 6 = 36 rectangles, we need to count only those which are squares.
- A square is formed when the distance between two horizontal lines is equal to the distance between two vertical lines.
- Since there are 4 horizontal and 4 vertical lines, there are 3 distances for each i.e. 3 horizontal distances and 3 vertical distances.
- Out of these 3 horizontal distances, only one distance is equal to each of the 3 vertical distances.
- Therefore, there are only 3 squares that can be formed by these 4 horizontal and 4 vertical lines.
- Hence, the total number of squares formed is 3.
Therefore, the maximum number of rectangles and squares that can be formed is 36 + 3 = 39, which is option C.
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