2019 AMC 10B Problems/Problem 5
Contents
Problem
Triangle lies in the first quadrant. Points
,
, and
are reflected across the line
to points
,
, and
, respectively. Assume that none of the vertices of the triangle lie on the line
. Which of the following statements is not always true?
Triangle
lies in the first quadrant.
Triangles
and
have the same area.
The slope of line
is
.
The slopes of lines
and
are the same.
Lines
and
are perpendicular to each other.
Solution
Let's 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.
Counterexamples
If and
, then the slope of
,
, is
, while the slope of
,
, is
.
is the
of
, but it is not the negative reciprocal of
. To generalize, let
denote the coordinates of point A, let
denote the coordinates of point B, let
denote the slope of segment
, and let
denote the slope of segment
. Then, the coordinate pair for
is
, and the pair for
is
. Then,
, and
. If
and
, $\frac{1}{m_{AB}} \neq \frac{1}{m_{A'B'}} \longRightarrow m_{AB} \neq m_{A'B'}$ (Error compiling LaTeX. Unknown error_msg), and in these cases, the condition is false.
~AlcBoy1729
See Also
2019 AMC 10B (Problems • Answer Key • Resources) | ||
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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