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Consider n x n graph paper, where n is a natural number. Consider the right-angled isosceles triangles, whose vertices are integer points of this graph and whose sides forming right angle are parallel to x and y axes. If the number of such triangle is 2/kn(n + 1)(2n + 1), the numerical quantity k must be equal to
Correct answer is '3'. Can you explain this answer?
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Most Upvoted Answer
Consider n x n graph paper, where n is a natural number. Consider the...
Explanation:
To find the value of k, we need to analyze the given expression and simplify it.
Analysis:
The given expression is:
2/kn(n-1)(2n-1)
We can break down this expression into four different factors:
1. 2
2. k
3. n
4. (n-1)(2n-1)
Let's analyze each factor:
Factor 1: 2
The factor 2 represents a constant value. It does not affect the value of k. Therefore, we can ignore it for now.
Factor 2: k
We need to determine the value of k that satisfies the given expression. As the question states that the number of right-angled isosceles triangles is equal to 2/kn(n-1)(2n-1), we can assume that k represents the number of right-angled isosceles triangles.
Factor 3: n
The factor n represents the size of the graph paper, which is an n x n grid. It is also a variable in the expression.
Factor 4: (n-1)(2n-1)
The factor (n-1)(2n-1) represents the number of lattice points on the graph paper. Each lattice point corresponds to a vertex of a right-angled isosceles triangle. To understand this, consider the following:
- (n-1) represents the number of lattice points on the x-axis, excluding the origin (0,0).
- (2n-1) represents the number of lattice points on the y-axis, excluding the origin (0,0).
- Multiplying these two factors gives us the total number of lattice points on the graph paper.
Simplification:
From the analysis, we have determined the significance of each factor. Now, let's simplify the expression by substituting the appropriate values:
Number of triangles = 2/kn(n-1)(2n-1)
Since the number of triangles is given as 2/kn(n-1)(2n-1), we can substitute this value for k:
2/kn(n-1)(2n-1) = 2/(2/n(n-1)(2n-1))
Simplifying further, we get:
2/(2/n(n-1)(2n-1)) = n(n-1)(2n-1)/2
Therefore, the value of k is equal to 2/n(n-1)(2n-1).
Conclusion:
From the simplified expression, we can see that the value of k is dependent on n. As there is no other information given in the question, we cannot determine a specific value for k. However, we can conclude that the value of k must be a function of n. Therefore, the correct answer cannot be determined based on the given information.
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Community Answer
Consider n x n graph paper, where n is a natural number. Consider the...
Let us first determine the number of possible squares on the graph.
The graph will have n × n squares of dimensions 1 x 1. (n - 1) × (n - 1) squares will rise to four isosceles right-angled triangles.
⇒ Required number of triangles
= 4.[n2 + (n - 1)2 + (n - 2)2 + ... + 12]
= 
⇒ k must be equal to 3.
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Consider n x n graph paper, where n is a natural number. Consider the right-angled isosceles triangles, whose vertices are integer points of this graph and whose sides forming right angle are parallel to x and y axes. If the number of such triangle is 2/kn(n + 1)(2n + 1), the numerical quantity k must be equal toCorrect answer is '3'. Can you explain this answer?
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