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Difference between revisions of "2023 AMC 10A Problems/Problem 19"

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==Problem==
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The line segment formed by <math>A(1, 2)</math> and <math>B(3, 3)</math> is rotated to the line segment formed by <math>A'(3, 1)</math> and <math>B'(4, 3)</math> about the point <math>P(r, s)</math>. What is <math>|r-s|</math>?
  
 +
<math>\textbf{(A) } \frac{1}{4} \qquad \textbf{(B) } \frac{1}{2} \qquad \textbf{(C) } \frac{3}{4}  \qquad \textbf{(D) } \frac{2}{3} \qquad  \textbf{(E) } 1</math>
 +
 +
==Solution 1==
 +
 +
Due to rotations preserving an equal distance, we can bash the answer with the distance formula. <math>D(A, P) = D(A', P)</math>, and <math>D(B, P) = D(B',P)</math>.
 +
Thus we will square our equations to yield:
 +
<math>(1-r)^2+(2-s)^2=(3-r)^2+(1-s)^2</math>, and <math>(3-r)^2+(3-s)^2=(4-r)^2+(3-s)^2</math>.
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Canceling <math>(3-s)^2</math> from the second equation makes it clear that <math>r</math> equals <math>3.5</math>.
 +
 +
Substituting will yield
 +
 +
<cmath>\begin{align*}(2.5)^2+(2-s)^2 &= (-0.5)^2+(1-s)^2 \\
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6.25+4-4s+s^2 &= 0.25+1-2s+s^2 \\
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2s &= 9 \\
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s &=4.5 \\
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\end{align*}
 +
</cmath>.
 +
 +
Now <math>|r-s| = |3.5-4.5| = \boxed{\textbf{(E) } 1}</math>.
 +
 +
-Antifreeze5420
 +
 +
==Solution 2==
 +
 +
Due to rotations preserving distance, we have that <math>BP = B^\prime P</math>, as well as <math>AP = A^\prime P</math>. From here, we can see that P must be on the perpendicular bisector of <math>\overline{BB^\prime}</math> due to the property of perpendicular bisectors keeping the distance to two points constant.
 +
 +
From here, we proceed to find the perpendicular bisector of <math>\overline{BB^\prime}</math>. We can see that this is just a horizontal line segment with midpoint at <math>(3.5, 3)</math>. This means that the equation of the perpendicular bisector is <math>x = 3.5</math>.
 +
 +
Similarly, we find the perpendicular bisector of <math>\overline{AA^\prime}</math>. We find the slope to be <math>\frac{1-2}{3-1} = -\frac12</math>, so our new slope will be <math>2</math>. The midpoint of <math>A</math> and <math>A^\prime</math> is <math>(2, \frac32)</math>, which we can use with our slope to get another equation of <math>y = 2x - \frac52</math>.
 +
 +
Now, point P has to lie on both of these perpendicular bisectors, meaning that it has to satisfy both equations. Plugging in the value of <math>x</math> we found earlier, we find that <math>y=4.5</math>. This means that <math>|r - s| = |3.5 - 4.5| = \boxed{\textbf{(E) } 1}</math>.
 +
 +
-DEVSAXENA
 +
 +
==Solution 3 (Coordinate Geometry)==
 +
 +
To find the center of rotation, we find the intersection point of the perpendicular bisectors of <math>\overline{AA^\prime}</math> and <math>\overline{BB^\prime}</math>.
 +
 +
We can find that the equation of the line <math>\overline{AA^\prime}</math> is <math>y = -\frac{1}{2}x + \frac{5}{2}</math>, and that the equation of the line <math>\overline{BB^\prime}</math> is <math>y = 3</math>.
 +
 +
When we solve for the perpendicular bisector of  <math>y = -\frac{1}{2}x + \frac{5}{2}</math>, we determine that it has a slope of 2, and it runs through <math>(2, 1.5)</math>. Plugging in <math>1.5 = 2(2)-n</math>, we get than <math>n = \frac{5}{2}</math>. Therefore our perpendicular bisector is <math>y=2x-\frac{5}{2}</math>. Next, we solve for the perpendicular of <math>y = 3</math>. We know that it has an undefined slope, and it runs through <math>(3.5, 3)</math>. We can determine that our second perpendicular bisector is <math>x = 3.5</math>.
 +
 +
Setting the equations equal to each other, we get <math>y = \frac{9}{2}</math>. Therefore, <math>|r - s| = |3.5 - 4.5| = \boxed{\textbf{(E) } 1}</math>.
 +
 +
<asy>
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pair A=(1,2);
 +
pair B=(3,3);
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pair A1=(3,1);
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pair B1=(4,3);
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dot("A",A,NW);
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dot("B",B,S);
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dot("A'",A1,S);
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dot("B'",B1,E);
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draw(A--A1);
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draw(B--B1);
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draw((3.5,0)--(3.5,6),BeginArrow(5),EndArrow(5));
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draw((1,-0.5)--(4,5.5),BeginArrow(5),EndArrow(5));
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pair P=(3.5,4.5);
 +
dot("P",P,NW);
 +
</asy>
 +
 +
~aydenlee & wuwang2002
 +
 +
==Solution 4==
 +
 +
We use the complex numbers approach to solve this problem.
 +
Denote by <math>\theta</math> the angle that <math>AB</math> rotates about <math>P</math> in the counterclockwise direction.
 +
 +
Thus, <math>A' - P = e^{i \theta} \left( A - P \right)</math> and <math>B' - P = e^{i \theta} \left( B - P \right)</math>.
 +
 +
Taking ratio of these two equations, we get
 +
<cmath>
 +
\[
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\frac{A' - P}{A - P} = \frac{B' - P}{B - P} .
 +
\]
 +
</cmath>
 +
 +
By solving this equation, we get <math>P = \frac{7}{2} + i \frac{9}{2}</math>.
 +
Therefore, <math>|s-t| = \left| \frac{7}{2} - \frac{9}{2} \right| = \boxed{\textbf{(E) 1}}</math>.
 +
 +
~Steven Chen (Professor Chen Education Palace, www.professorchenedu.com)
 +
 +
== Video Solution ==
 +
https://youtu.be/aYHwBpdWkAk
 +
 +
== Video Solution by ==
 +
 +
https://www.youtube.com/watch?v=fIzCR4x4x-M
 +
 +
== Video Solution 1 by OmegaLearn ==
 +
https://youtu.be/88F18qth0xI
 +
 +
==Video Solution by SpreadTheMathLove==
 +
https://www.youtube.com/watch?v=oHMJwiEwOS0
 +
 +
==Video Solution==
 +
 +
https://youtu.be/va3ZCFeKfzg
 +
 +
~Steven Chen (Professor Chen Education Palace, www.professorchenedu.com)
 +
== Video Solution by TheBeautyofMath==
 +
https://youtu.be/Q_4uMxMbQlI
 +
 +
~IceMatrix
 +
==See Also==
 +
{{AMC10 box|year=2023|ab=A|num-b=18|num-a=20}}
 +
{{MAA Notice}}

Revision as of 21:21, 6 August 2024

Problem

The line segment formed by $A(1, 2)$ and $B(3, 3)$ is rotated to the line segment formed by $A'(3, 1)$ and $B'(4, 3)$ about the point $P(r, s)$. What is $|r-s|$?

$\textbf{(A) } \frac{1}{4} \qquad \textbf{(B) } \frac{1}{2} \qquad \textbf{(C) } \frac{3}{4} \qquad \textbf{(D) } \frac{2}{3} \qquad \textbf{(E) } 1$

Solution 1

Due to rotations preserving an equal distance, we can bash the answer with the distance formula. $D(A, P) = D(A', P)$, and $D(B, P) = D(B',P)$. Thus we will square our equations to yield: $(1-r)^2+(2-s)^2=(3-r)^2+(1-s)^2$, and $(3-r)^2+(3-s)^2=(4-r)^2+(3-s)^2$. Canceling $(3-s)^2$ from the second equation makes it clear that $r$ equals $3.5$.

Substituting will yield

\begin{align*}(2.5)^2+(2-s)^2 &= (-0.5)^2+(1-s)^2 \\ 6.25+4-4s+s^2 &= 0.25+1-2s+s^2 \\ 2s &= 9 \\ s &=4.5 \\ \end{align*}.

Now $|r-s| = |3.5-4.5| = \boxed{\textbf{(E) } 1}$.

-Antifreeze5420

Solution 2

Due to rotations preserving distance, we have that $BP = B^\prime P$, as well as $AP = A^\prime P$. From here, we can see that P must be on the perpendicular bisector of $\overline{BB^\prime}$ due to the property of perpendicular bisectors keeping the distance to two points constant.

From here, we proceed to find the perpendicular bisector of $\overline{BB^\prime}$. We can see that this is just a horizontal line segment with midpoint at $(3.5, 3)$. This means that the equation of the perpendicular bisector is $x = 3.5$.

Similarly, we find the perpendicular bisector of $\overline{AA^\prime}$. We find the slope to be $\frac{1-2}{3-1} = -\frac12$, so our new slope will be $2$. The midpoint of $A$ and $A^\prime$ is $(2, \frac32)$, which we can use with our slope to get another equation of $y = 2x - \frac52$.

Now, point P has to lie on both of these perpendicular bisectors, meaning that it has to satisfy both equations. Plugging in the value of $x$ we found earlier, we find that $y=4.5$. This means that $|r - s| = |3.5 - 4.5| = \boxed{\textbf{(E) } 1}$.

-DEVSAXENA

Solution 3 (Coordinate Geometry)

To find the center of rotation, we find the intersection point of the perpendicular bisectors of $\overline{AA^\prime}$ and $\overline{BB^\prime}$.

We can find that the equation of the line $\overline{AA^\prime}$ is $y = -\frac{1}{2}x + \frac{5}{2}$, and that the equation of the line $\overline{BB^\prime}$ is $y = 3$.

When we solve for the perpendicular bisector of $y = -\frac{1}{2}x + \frac{5}{2}$, we determine that it has a slope of 2, and it runs through $(2, 1.5)$. Plugging in $1.5 = 2(2)-n$, we get than $n = \frac{5}{2}$. Therefore our perpendicular bisector is $y=2x-\frac{5}{2}$. Next, we solve for the perpendicular of $y = 3$. We know that it has an undefined slope, and it runs through $(3.5, 3)$. We can determine that our second perpendicular bisector is $x = 3.5$.

Setting the equations equal to each other, we get $y = \frac{9}{2}$. Therefore, $|r - s| = |3.5 - 4.5| = \boxed{\textbf{(E) } 1}$.

[asy] pair A=(1,2); pair B=(3,3); pair A1=(3,1); pair B1=(4,3); dot("A",A,NW); dot("B",B,S); dot("A'",A1,S); dot("B'",B1,E); draw(A--A1); draw(B--B1); draw((3.5,0)--(3.5,6),BeginArrow(5),EndArrow(5)); draw((1,-0.5)--(4,5.5),BeginArrow(5),EndArrow(5)); pair P=(3.5,4.5); dot("P",P,NW); [/asy]

~aydenlee & wuwang2002

Solution 4

We use the complex numbers approach to solve this problem. Denote by $\theta$ the angle that $AB$ rotates about $P$ in the counterclockwise direction.

Thus, $A' - P = e^{i \theta} \left( A - P \right)$ and $B' - P = e^{i \theta} \left( B - P \right)$.

Taking ratio of these two equations, we get \[ \frac{A' - P}{A - P} = \frac{B' - P}{B - P} . \]

By solving this equation, we get $P = \frac{7}{2} + i \frac{9}{2}$. Therefore, $|s-t| = \left| \frac{7}{2} - \frac{9}{2} \right| = \boxed{\textbf{(E) 1}}$.

~Steven Chen (Professor Chen Education Palace, www.professorchenedu.com)

Video Solution

https://youtu.be/aYHwBpdWkAk

Video Solution by

https://www.youtube.com/watch?v=fIzCR4x4x-M

Video Solution 1 by OmegaLearn

https://youtu.be/88F18qth0xI

Video Solution by SpreadTheMathLove

https://www.youtube.com/watch?v=oHMJwiEwOS0

Video Solution

https://youtu.be/va3ZCFeKfzg

~Steven Chen (Professor Chen Education Palace, www.professorchenedu.com)

Video Solution by TheBeautyofMath

https://youtu.be/Q_4uMxMbQlI

~IceMatrix

See Also

2023 AMC 10A (Problems • Answer Key • Resources)
Preceded by
Problem 18 		Followed by
Problem 20
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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