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Bunuel
The two adjacent vertices of a quadrilateral are (2, 3), (8, 3). If the third and the fourth vertices of the quadrilateral are (x, y) and (a, b), for how many pairs of non-negative integer values of (x, y) and (a, b), the quadrilateral will be a rectangle of area at least 12 sq. units and at most 30 sq. units?

A. 2
B. 4
C. 5
D. 6
E. >6

Solution:

  • Two adjacent vertices are (2, 3) and (8, 3). The length between them is 6 units.
  • This means this side (of length 6) is parallel to the x-axis
  • To form rectangles, other sides will have to be parallel to the x-axis and y-axes
  • Possible value of width are
    • Width = 2 (2 ways - (2, 5),(8, 5) and (2, 1),(8, 1))
    • Width = 3 (2 ways - (2, 6),(8, 6) and (2, 0),(8, 0))
    • Width = 4 (1 way - (2, 7),(8, 7))
    • Width = 5 (1 way - (2, 8),(8, 8))
  • Thus, total 6 pairs possible

Hence the right answer is Option D
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