2023 AMC 8

Complete problem set with solutions and individual problem pages

Problem 7 Easy

A rectangle, with sides parallel to the x-axis and y-axis, has opposite vertices located at (15, 3) and (16, 5). A line is drawn through points A(0, 0) and B(3, 1). Another line is drawn through points C(0, 10) and D(2, 9). How many points on the rectangle lie on at least one of the two lines?

  • A.

    0

  • B.

    1

  • C.

    2

  • D.

    3

  • E.

    4

Answer:B

Solution 1

If we extend the lines, we have the following diagram:

Therefore, we see that the answer is \boxed{\textbf{(B)}\ 1}.

 

Solution 2

Note that the y-intercepts of line AB and line CD are 0 and 10. If the analytic expression for line AB is y=k_{1}x, and the analytic expression for line CD is y=k_{2}x+10, we have equations: 3k_{1} = 1 and 2k_{2} + 10 = 9. Solving these equations, we can find out that k_{1} = \frac{1}{3} and k_{2} = -\frac{1}{2}. Therefore, we can determine that the expression for line AB is y=\frac{1}{3}x and the expression for line CD is y=-\frac{1}{2}x + 10. When x=15, the coordinates that line AB and line CD pass through are (15, 5) and \left(15, \frac{5}{2}\right), and (15, 5) lies perfectly on one vertex of the rectangle while the y coordinate of \left(15, \frac{5}{2}\right) is out of the range 3 \leq y \leq 5 (lower than the bottom left corner of the rectangle (15, 3)). Considering that the y value of the line CD will only decrease, and the y value of the line AB will only increase, there will not be another point on the rectangle that lies on either of the two lines. Thus, we can conclude that the answer is \boxed{\textbf{(B)}\ 1}.

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