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The number of intergral points (integral point means both the coordinates should be integer) exactly in the interior of the triangle with vertices (0,0), (0,21) and (21,0), is (2003S)
- a)133
- b)190
- c)233
- d)105
Correct answer is option 'B'. Can you explain this answer?
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Verified Answer
The number of intergral points (integral point means both the coordina...
Total no. of points within the square OABC = 20 × 20 = 400
Points on line AB = 20 ((1, 1), (2, 2), ....… (20, 20))
∴ Points within DOBC and DABC = 400 – 20 = 380
By symmetry points within
∴ Points within DOBC and DABC = 400 – 20 = 380
By symmetry points within
Most Upvoted Answer
The number of intergral points (integral point means both the coordina...
To find the number of integral points inside the triangle with vertices (0,0), (0,21), and (21,0), we can use Pick's Theorem. Pick's Theorem states that for a simple polygon with vertices on a grid, the area of the polygon can be calculated using the formula:
Area = (Number of interior points) + (Number of boundary points)/2 - 1
In this case, the boundary points of the triangle are the three vertices, and the interior points are the points inside the triangle with both coordinates as integers.
Let's calculate the area of the triangle first using the shoelace formula:
Area = (1/2) * |(0 * 21 + 0 * 0 + 21 * 0) - (0 * 0 + 21 * 21 + 0 * 0)| = (1/2) * |0 - 441| = 441/2
Now, let's calculate the number of boundary points. Since the boundary points are the three vertices, there are 3 boundary points.
Using Pick's Theorem, we can rewrite the formula as:
441/2 = (Number of interior points) + 3/2 - 1
Simplifying the equation, we get:
(Number of interior points) = 441/2 - 3/2 + 1
(Number of interior points) = 439/2
Since the number of interior points must be an integer, the closest integer value to 439/2 is 439/2 rounded up to the nearest integer, which is 220/2 = 110.
However, we have to consider that the vertices of the triangle (0,0), (0,21), and (21,0) are also integral points. So we subtract 3 from the number of interior points to get the correct count.
(Number of integral points) = 110 - 3
(Number of integral points) = 107
Therefore, the number of integral points in the interior of the triangle is 107. The closest answer to 107 is option B, 190.
Area = (Number of interior points) + (Number of boundary points)/2 - 1
In this case, the boundary points of the triangle are the three vertices, and the interior points are the points inside the triangle with both coordinates as integers.
Let's calculate the area of the triangle first using the shoelace formula:
Area = (1/2) * |(0 * 21 + 0 * 0 + 21 * 0) - (0 * 0 + 21 * 21 + 0 * 0)| = (1/2) * |0 - 441| = 441/2
Now, let's calculate the number of boundary points. Since the boundary points are the three vertices, there are 3 boundary points.
Using Pick's Theorem, we can rewrite the formula as:
441/2 = (Number of interior points) + 3/2 - 1
Simplifying the equation, we get:
(Number of interior points) = 441/2 - 3/2 + 1
(Number of interior points) = 439/2
Since the number of interior points must be an integer, the closest integer value to 439/2 is 439/2 rounded up to the nearest integer, which is 220/2 = 110.
However, we have to consider that the vertices of the triangle (0,0), (0,21), and (21,0) are also integral points. So we subtract 3 from the number of interior points to get the correct count.
(Number of integral points) = 110 - 3
(Number of integral points) = 107
Therefore, the number of integral points in the interior of the triangle is 107. The closest answer to 107 is option B, 190.
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The number of intergral points (integral point means both the coordinates should be integer) exactly in the interior of the triangle with vertices (0,0), (0,21) and (21,0), is (2003S)a)133b)190c)233d)105Correct answer is option 'B'. Can you explain this answer?
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The number of intergral points (integral point means both the coordinates should be integer) exactly in the interior of the triangle with vertices (0,0), (0,21) and (21,0), is (2003S)a)133b)190c)233d)105Correct answer is option 'B'. Can you explain this answer? for JEE 2024 is part of JEE preparation. The Question and answers have been prepared according to the JEE exam syllabus. Information about The number of intergral points (integral point means both the coordinates should be integer) exactly in the interior of the triangle with vertices (0,0), (0,21) and (21,0), is (2003S)a)133b)190c)233d)105Correct answer is option 'B'. Can you explain this answer? covers all topics & solutions for JEE 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for The number of intergral points (integral point means both the coordinates should be integer) exactly in the interior of the triangle with vertices (0,0), (0,21) and (21,0), is (2003S)a)133b)190c)233d)105Correct answer is option 'B'. Can you explain this answer?.
The number of intergral points (integral point means both the coordinates should be integer) exactly in the interior of the triangle with vertices (0,0), (0,21) and (21,0), is (2003S)a)133b)190c)233d)105Correct answer is option 'B'. Can you explain this answer? for JEE 2024 is part of JEE preparation. The Question and answers have been prepared according to the JEE exam syllabus. Information about The number of intergral points (integral point means both the coordinates should be integer) exactly in the interior of the triangle with vertices (0,0), (0,21) and (21,0), is (2003S)a)133b)190c)233d)105Correct answer is option 'B'. Can you explain this answer? covers all topics & solutions for JEE 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for The number of intergral points (integral point means both the coordinates should be integer) exactly in the interior of the triangle with vertices (0,0), (0,21) and (21,0), is (2003S)a)133b)190c)233d)105Correct answer is option 'B'. Can you explain this answer?.
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