Difference between revisions of "2017 AMC 10A Problems/Problem 12"

Revision as of 13:20, 8 June 2021

Problem

Let $S$ be a set of points $(x,y)$ in the coordinate plane such that two of the three quantities $3,~x+2,$ and $y-4$ are equal and the third of the three quantities is no greater than this common value. Which of the following is a correct description for $S?$

$\textbf{(A)}\ \text{a single point} \qquad\textbf{(B)}\ \text{two intersecting lines} \\\qquad\textbf{(C)}\ \text{three lines whose pairwise intersections are three distinct points} \\\qquad\textbf{(D)}\ \text{a triangle} \qquad\textbf{(E)}\ \text{three rays with a common endpoint}$

Diagram

~BakedPotato66

Solution

If the two equal values are $3$ and $x+2$ , then $x=1$ . Also, $y-4\le 3$ because $3$ is the common value. Solving for $y$ , we get $y \le 7$ . Therefore the portion of the line $x=1$ where $y \le 7$ is part of $S$ . This is a ray with an endpoint of $(1, 7)$ .

Similar to the process above, we assume that the two equal values are $3$ and $y-4$ . Solving the equation $3=y-4$ then $y=7$ . Also, $x+2\le 3$ because 3 is the common value. Solving for $x$ , we get $x\le1$ . Therefore the portion of the line $y=7$ where $x\le 1$ is also part of $S$ . This is another ray with the same endpoint as the above ray: $(1, 7)$ .

If $x+2$ and $y-4$ are the two equal values, then $x+2=y-4$ . Solving the equation for $y$ , we get $y=x+6$ . Also $3\le y-4$ because $y-4$ is one way to express the common value. Solving for $y$ , we get $y\ge 7$ . Therefore the portion of the line $y=x+6$ where $y\ge 7$ is part of $S$ like the other two rays. The lowest possible value that can be achieved is also $(1, 7)$ .

Since $S$ is made up of three rays with common endpoint $(1, 7)$ , the answer is $\boxed{\textbf{(E) }\text{three rays with a common endpoint}}$

Video Solution

https://youtu.be/s4vnGlwwHHw?t=190

@@ Line 5: / Line 5: @@
 <math>\textbf{(A)}\ \text{a single point} \qquad\textbf{(B)}\ \text{two intersecting lines} \\\qquad\textbf{(C)}\ \text{three lines whose pairwise intersections are three distinct points} \\\qquad\textbf{(D)}\ \text{a triangle} \qquad\textbf{(E)}\ \text{three rays with a common endpoint}</math>
+==Diagram==
+[[File:BakedPotato66 (Desmos Graph).png|900px|caption]]
+~BakedPotato66
 ==Solution==
 If the two equal values are <math>3</math> and <math>x+2</math>, then <math>x=1</math>. Also, <math>y-4\le 3</math> because <math>3</math> is the common value. Solving for <math>y</math>, we get <math>y \le 7</math>. Therefore the portion of the line <math>x=1</math> where <math>y \le 7</math> is part of <math>S</math>. This is a ray with an endpoint of <math>(1, 7)</math>.

2017 AMC 10A (Problems • Answer Key • Resources)
Preceded by Problem 11	Followed by Problem 13
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