Difference between revisions of "2019 AMC 10B Problems/Problem 5"
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− | + | ==Problem== | |
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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{(B) } </math> Triangles <math>ABC</math> and <math>A'B'C'</math> have the same area. | ||
+ | |||
+ | <math>\textbf{(C) } </math> The slope of line <math>AA'</math> is <math>-1</math>. | ||
+ | |||
+ | <math>\textbf{(D) } </math> The slopes of lines <math>AA'</math> and <math>CC'</math> are the same. | ||
+ | |||
+ | <math>\textbf{(E) } </math> Lines <math>AB</math> and <math>A'B'</math> are perpendicular to each other. | ||
+ | |||
+ | ==Solution== | ||
+ | Let's analyze all of the options separately. | ||
+ | |||
+ | <math>\textbf{(A)}</math>: Clearly <math>\textbf{(A)}</math> is true, because a point in the first quadrant will have non-negative <math>x</math>- and <math>y</math>-coordinates, and so its reflection, with the coordinates swapped, will also have non-negative <math>x</math>- and <math>y</math>-coordinates. | ||
+ | |||
+ | <math>\textbf{(B)}</math>: The triangles have the same area, since <math>\triangle ABC</math> and <math>\triangle A'B'C'</math> are the same triangle (congruent). More formally, we can say that area is ''invariant'' under reflection. | ||
+ | |||
+ | <math>\textbf{(C)}</math>: If point <math>A</math> has coordinates <math>(p,q)</math>, then <math>A'</math> will have coordinates <math>(q,p)</math>. The gradient is thus <math>\frac{p-q}{q-p} = -1</math>, so this is true. (We know <math>p \neq q</math> since the question states that none of the points <math>A</math>, <math>B</math>, or <math>C</math> lies on the line <math>y=x</math>, so there is no risk of division by zero). | ||
+ | |||
+ | <math>\textbf{(D)}</math>: Repeating the argument for <math>\textbf{(C)}</math>, we see that both lines have slope <math>-1</math>, so this is also true. | ||
+ | |||
+ | <math>\textbf{(E)}</math>: This is the only one left, presumably the answer. To prove: if point <math>A</math> has coordinates <math>(p,q)</math> and point <math>B</math> has coordinates <math>(r,s)</math>, then <math>A'</math> and <math>B'</math> will, respectively, have coordinates <math>(q,p)</math> and <math>(s,r)</math>. The product of the gradients of <math>AB</math> and <math>A'B'</math> is <math>\frac{s-q}{r-p} \cdot \frac{r-p}{s-q} = 1 \neq -1</math>, so in fact these lines are '''never''' perpendicular to each other (using the "negative reciprocal" condition for perpendicularity). | ||
+ | |||
+ | Thus the answer is <math>\boxed{\textbf{(E)}}</math>. | ||
+ | |||
+ | ==Counterexamples== | ||
+ | If <math>(x_1,y_1) = (2,3)</math> and <math>(x_2,y_2) = (7,1)</math>, then the slope of <math>AB</math>, <math>m_{AB}</math>, is <math>\frac{1 - 3}{7 - 2} = -\frac{2}{5}</math>, while the slope of <math>A'B'</math>, <math>m_{A'B'}</math>, is <math>\frac{7 - 2}{1 - 3} = -\frac{5}{2}</math>. <math>m_{A'B'}</math> is the '''reciprocal''' of <math>m_{AB}</math>, but it is not the negative reciprocal of <math>m_{AB}</math>. To generalize, let <math>(x_1,y_1)</math> denote the coordinates of point <math>A</math>, let <math>(x_2, y_2)</math> denote the coordinates of point <math>B</math>, let <math>m_{AB}</math> denote the slope of segment <math>\overline{AB}</math>, and let <math>m_{A'B'}</math> denote the slope of segment <math>\overline{A'B'}</math>. Then, the coordinates of <math>A'</math> are <math>(y_1, x_1)</math>, and of <math>B'</math> are <math>(y_2, x_2)</math>. Then, <math>m_{AB} = \frac{y_2 - y_1}{x_2 - x_1}</math>, and <math>m_{A'B'} = \frac{x_2 - x_1}{y_2 - y_1} = \frac{1}{m_{AB}}</math>, so slopes arent negative reciprocals of each other. | ||
+ | |||
+ | ==Video Solution== | ||
+ | https://youtu.be/XKSZ9o54dg8 | ||
+ | |||
+ | ~Education, the Study of Everything | ||
+ | |||
+ | ==Video Solution== | ||
+ | https://youtu.be/dYn6jYRgEv4 | ||
+ | |||
+ | ~savannahsolver | ||
+ | |||
+ | ==See Also== | ||
+ | |||
+ | {{AMC10 box|year=2019|ab=B|num-b=4|num-a=6}} | ||
+ | {{MAA Notice}} |
Latest revision as of 21:24, 5 October 2023
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.
: Clearly
is true, because a point in the first quadrant will have non-negative
- and
-coordinates, and so its reflection, with the coordinates swapped, will also have non-negative
- and
-coordinates.
: The triangles have the same area, since
and
are the same triangle (congruent). More formally, we can say that area is invariant under reflection.
: If point
has coordinates
, then
will have coordinates
. The gradient is thus
, so this is true. (We know
since the question states that none of the points
,
, or
lies on the line
, so there is no risk of division by zero).
: Repeating the argument for
, we see that both lines have slope
, so this is also true.
: This is the only one left, presumably the answer. To prove: if point
has coordinates
and point
has coordinates
, then
and
will, respectively, have coordinates
and
. The product of the gradients of
and
is
, so in fact these lines are never perpendicular to each other (using the "negative reciprocal" condition for perpendicularity).
Thus the answer is .
Counterexamples
If and
, then the slope of
,
, is
, while the slope of
,
, is
.
is the reciprocal of
, but it is not the negative reciprocal of
. To generalize, let
denote the coordinates of point
, let
denote the coordinates of point
, let
denote the slope of segment
, and let
denote the slope of segment
. Then, the coordinates of
are
, and of
are
. Then,
, and
, so slopes arent negative reciprocals of each other.
Video Solution
~Education, the Study of Everything
Video Solution
~savannahsolver
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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