Difference between revisions of "2019 AMC 10B Problems/Problem 5"

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==Problem==
 
==Problem==
  
Triangle <math>ABC</math> lies in the first quadrant. Points <math>A</math>, <math>B</math>, and <math>C</math> are reflected across the line <math>y=x</math> to points <math>A'</math>, <math>B'</math>, and <math>C'</math>, respectively. Assume that none of the vertices of the triangle lie on the line <math>y=x</math>. Which of the following statements is not always true?
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Triangle <math>ABC</math> lies in the first quadrant. Points <math>A</math>, <math>B</math>, and <math>C</math> are reflected across the line <math>y=x</math> to points <math>A'</math>, <math>B'</math>, and <math>C'</math>, respectively. Assume that none of the vertices of the triangle lie on the line <math>y=x</math>. Which of the following statements is <i><u>not</u></i> always true?
  
 
<math>\textbf{(A) } </math> Triangle <math>A'B'C'</math> lies in the first quadrant.
 
<math>\textbf{(A) } </math> Triangle <math>A'B'C'</math> lies in the first quadrant.

Revision as of 18:14, 14 February 2019

Problem

Triangle $ABC$ lies in the first quadrant. Points $A$, $B$, and $C$ are reflected across the line $y=x$ to points $A'$, $B'$, and $C'$, respectively. Assume that none of the vertices of the triangle lie on the line $y=x$. Which of the following statements is not always true?

$\textbf{(A) }$ Triangle $A'B'C'$ lies in the first quadrant.

$\textbf{(B) }$ Triangles $ABC$ and $A'B'C'$ have the same area.

$\textbf{(C) }$ The slope of line $AA'$ is $-1$.

$\textbf{(D) }$ The slopes of lines $AA'$ and $CC'$ are the same.

$\textbf{(E) }$ Lines $AB$ and $A'B'$ are perpendicular to each other.

Solution

Lets analyze all of the options separately. A: Clearly A is true, because a coordinate in the first quadrant will have (+,+), and its inverse would also have (+,+) B: The triangles have the same area, it's the same triangle. C: If coordinate A has (x,y), then its inverse will have (y,x). (x-y)/(y-x)=-1, so this is true. D: Likewise, if coordinate A has (x1,y1), and AA' has a slope of -1, then coordinate B, with (x2,y2), will also have a slope of -1. This is true. E: By process of elimination, this is the answer, but if coordinate A has (x1,y1) and coordinate B has (x2,y2), then their inverses will be (y1,x1), (y2,x2), and it is not necessarily true that (y2-y1)/(x2-x1)=-(y2-y1)/(x2-x1). (Negative inverses of each other). Clearly, the answer is E.

See Also

2019 AMC 10B (ProblemsAnswer KeyResources)
Preceded by
Problem 4
Followed by
Problem 6
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 AMC 10 Problems and Solutions

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