Difference between revisions of "2023 AMC 8 Problems/Problem 7"

(Solution 2)
(Solution 2)
Line 17: Line 17:
 
Note that the <math> y </math>-intercept of line <math> AB </math> and line <math> CD </math> are <math> 0 </math> and <math> 10 </math>. If the analytic expression for line <math> AB </math> is <math> y=k_{1}x </math>, and the analytic expression for line <math> CD </math> is <math> y=k_{2}x+10 </math>, we have equations:<math> 3k_{1} = 1 </math> and <math> 2k_{2} + 10 = 9 </math>. Solving these equations, we can find out that <math> k_{1} = \frac{1}{3} </math> and <math> k_{2} = -\frac{1}{2} </math>. Therefore, we can determine that the expression for line <math> AB </math> is <math> y=\frac{1}{3}x </math>. and the expression for line <math> CD </math> is <math> y=-\frac{1}{2}x + 10 </math>. When <math> x=15 </math>, the coordinates that line <math> AB </math> and line <math> CD </math> pass through are <math> (15, 5) </math> and <math> \left(15, \frac{5}{2}\right) </math>, and <math> (15, 5) </math> lies perfectly on one vertex of the rectangle while the <math> y </math> coordinate of <math> \left(15, \frac{5}{2}\right) </math> is out of the range <math> 3 \leq y \leq 5 </math> (lower than the bottom left corner of the rectangle <math> (15, 3) </math>). Considering that the <math> y </math> value of the line <math> CD </math> will only decrease, and the <math> y </math> value of the line <math> AB </math> 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 <math>\boxed{\textbf{(B)}\ 1}</math>
 
Note that the <math> y </math>-intercept of line <math> AB </math> and line <math> CD </math> are <math> 0 </math> and <math> 10 </math>. If the analytic expression for line <math> AB </math> is <math> y=k_{1}x </math>, and the analytic expression for line <math> CD </math> is <math> y=k_{2}x+10 </math>, we have equations:<math> 3k_{1} = 1 </math> and <math> 2k_{2} + 10 = 9 </math>. Solving these equations, we can find out that <math> k_{1} = \frac{1}{3} </math> and <math> k_{2} = -\frac{1}{2} </math>. Therefore, we can determine that the expression for line <math> AB </math> is <math> y=\frac{1}{3}x </math>. and the expression for line <math> CD </math> is <math> y=-\frac{1}{2}x + 10 </math>. When <math> x=15 </math>, the coordinates that line <math> AB </math> and line <math> CD </math> pass through are <math> (15, 5) </math> and <math> \left(15, \frac{5}{2}\right) </math>, and <math> (15, 5) </math> lies perfectly on one vertex of the rectangle while the <math> y </math> coordinate of <math> \left(15, \frac{5}{2}\right) </math> is out of the range <math> 3 \leq y \leq 5 </math> (lower than the bottom left corner of the rectangle <math> (15, 3) </math>). Considering that the <math> y </math> value of the line <math> CD </math> will only decrease, and the <math> y </math> value of the line <math> AB </math> 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 <math>\boxed{\textbf{(B)}\ 1}</math>
  
~Bloggish
+
~\href{https://artofproblemsolving.com/wiki/index.php/User:Bloggish}{Bloggish}
  
 
==Video Solution by Magic Square==
 
==Video Solution by Magic Square==

Revision as of 22:02, 26 January 2023

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

Note that the $y$-intercept 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 $\boxed{\textbf{(B)}\ 1}$

~\href{https://artofproblemsolving.com/wiki/index.php/User:Bloggish}{Bloggish}

Video Solution by Magic Square

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

See Also

2023 AMC 8 (ProblemsAnswer KeyResources)
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