AMC 8 2023 Problem 7

AMC8 2023 年第 7 题

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

有一个矩形，它的边平行于 x 轴和 y轴，两个相对的顶点分别是（15，3）与（16，5）。过点 A(0,0) 与 B(3,1) 作一条直线，再过点 C(0,10) 与 D(2,9) 作另一条直线。问矩形的边界上有多少个点落在两条直线中至少一条上？

正确答案 · Correct Answer

解析 · Solution

Solution 1

Note that the

y

-intercepts of line

AB

and line

CD

are 0 and 10. If the analytic expression for line

AB

y=k_{1}x

, and the analytic expression for line

CD

y=k_{2}x+10

, we have equations:

3k_{1} = 1

and

2k_{2} + 10 = 9

. Solving these equations, we can find out that

\displaystyle k_{1} = \frac{1}{3}

and

\displaystyle k_{2} = - \frac{1}{2}

. Therefore, we can determine that the expression for line

AB

\displaystyle y= \frac{1}{3} x

and the expression for line

CD

\displaystyle y=- \frac{1}{2} x + 10

. When x=15, the coordinates that line

AB

and line

CD

pass through are (15, 5) and

\displaystyle \left(15, \frac{5}{2} \right)

, and (15, 5) lies perfectly on one vertex of the rectangle while the

y

coordinate of

\displaystyle \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 1.

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