GATE ME 2015 Set 3 — Question 9
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General Aptitude → Quantitative Aptitude → Mensuration & Geometry
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Question
Right triangle is to be constructed in the - plane so that the right angle is at and line is parallel to the -axis. The and coordinates of are to be integers that satisfy the inequalities: and . How many different triangles could be constructed with these properties?
Correct answer
(C) 9,900
Solution
To construct the triangle with the given properties:
Vertex can be any of the 110 available points. Let .
Number of choices for = .
Number of choices for = .
Total = .
1.Identify the range of coordinates:
- For : . The number of integer values is .
- For : . The number of integer values is .
- Total available integer points in the region = .
Vertex can be any of the 110 available points. Let .
3.Choose vertex :
Line must be parallel to the -axis, so . Vertex must have an -coordinate from the 10 available values. Number of choices for = .
4.Choose vertex :
Since the right angle is at , must be perpendicular to . Since is horizontal, must be vertical. Thus, . Vertex must have a -coordinate from the 11 available values.Number of choices for = .
5.Calculate total triangles:
Total triangles = (Choices for ) (Choices for ) (Choices for )Total = .
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