Art of Problem Solving
AoPS Online
Math texts, online classes, and more
for students in grades 5-12.
Visit AoPS Online ‚
Books for Grades 5-12 Online Courses
Beast Academy
Engaging math books and online learning
for students ages 6-13.
Visit Beast Academy ‚
Books for Ages 6-13 Beast Academy Online
AoPS Academy
Small live classes for advanced math
and language arts learners in grades 2-12.
Visit AoPS Academy ‚
Find a Physical Campus Visit the Virtual Campus

2023 AMC 8 Problems/Problem 7

Revision as of 06:23, 26 January 2023 by Bloggish (talk | contribs) (→‎Solution 2)

Problem

A rectangle, with sides parallel to the $x$-axis and $y$-axis, has opposite vertices located at $(15, 3)$ and $(16, 5)$. A line 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?

$\textbf{(A)}\ 0 \qquad \textbf{(B)}\ 1 \qquad \textbf{(C)}\ 2 \qquad \textbf{(D)}\ 3 \qquad \textbf{(E)}\ 4$

Solution 1

If we extend the lines, we have

Screenshot 2023-01-25 8.26.48 AM.png

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

~MrThinker

Solution 2

If the analytic expression for line $AB$ is $y=k_{1}x+b_{1}$, and the analytic expression for line $CD$ is $y=k_{2}x+b_{2}$, we have the systems of equations: \[\begin{cases} b_{1} = 0 \\ 3k_{1} + b_{1} = 1 \end{cases}\] and \[\begin{cases} b_{2} = 10 \\ 2k_{2} + b_{2} = 9 \end{cases}\] Solving the systems, we have: \[\begin{cases} k_{1} = \frac{1}{3} \\ b_{1} = 0 \end{cases}\] and \[\begin{cases} k_{2} = -\frac{1}{2} \\ b_{2} = 10 \end{cases}\] 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 $(15, \frac{5}{2})$, and $(15, 5)$ lies perfectly on one vertex of the rectangle while the $y$ coordinate of $(15, \frac{5}{2})$ 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 $\boxed{\textbf{(B)}\ 1}$

~Bloggish

Video Solution by Magic Square

https://youtu.be/-N46BeEKaCQ?t=5151

See Also

2023 AMC 8 (Problems • Answer Key • Resources)
Preceded by
Problem 6 		Followed by
Problem 8
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 • 16 • 17 • 18 • 19 • 20 • 21 • 22 • 23 • 24 • 25
All AJHSME/AMC 8 Problems and Solutions

The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions. AMC logo.png

Retrieved from "https://artofproblemsolving.com/wiki/index.php?title=2023_AMC_8_Problems/Problem_7&oldid=188051"